Catalog

All 730 registered systems, by field and by tag. Select one by its name with create_system or create_simulation of skcomplex.systems, and read what it needs with describe. The search box of the site finds a system by its name, description, or tag.

from skcomplex.systems import catalog, create_simulation, create_system, describe

names = [entry.name for entry in catalog(field='biology/ecology')]
entry = describe('lotka_volterra')
system = create_system('lotka_volterra')

By field

biology/collective_behaviour

• Active walkers and trail formation active_walkers: Active walkers who walk between the corners of a green and wear a system of trails into it.
• Ant colony optimization on a road network ant_colony_optimization: Ants that find the shortest path between two points of a Delaunay road network, round after round.
• Binary ant decision binary_ant_decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Bird swarm with pair potentials bird_swarm: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions boids: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Calm and nervous rats rats: A colony of calm and nervous rats, in which nervousness spreads by contact and fades on its own.
• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.
• Pheromone foraging pheromone_foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Strömbom shepherding of a flock shepherding: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.

biology/ecology

• Agent-based Lotka-Volterra agent_based_lotka_volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.
• Competition competition: Two species that compete for the same resources, which coexist or exclude each other.
• Consumer-resource models of substitutable and essential resources consumer_resource: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.
• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Exponential growth exponential_growth: A population that grows or declines at a constant per-capita rate.
• Fibonacci rabbits fibonacci_rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.
• Fisher-KPP fisher_kpp: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).
• Food webs food_webs: A food web where basal species grow logistically and consumers live on Lotka-Volterra predation.
• Generalized Lotka-Volterra microbial community generalized_lotka_volterra: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).
• Generalized stochastic predator-prey stochastic_predator_prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.
• Hutchinson hutchinson: Hutchinson's delayed logistic equation, which oscillates around K when r * tau exceeds π/2.
• Intrinsic-instability populations intrinsic_instability: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.
• Logistic growth logistic_growth: A population that grows logistically towards its carrying capacity.
• Logistic growth with a varying capacity varying_capacity_logistic: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.
• Lotka-Volterra lotka_volterra: Prey and predators whose populations oscillate through predation.
• Lotka-Volterra with harvesting lotka_volterra_with_harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Mutualism mutualism: Two species that grow logistically and feed each other, above their carrying capacities.
• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Neutral macroecology neutral_macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.
• Nicholson-Bailey nicholson_bailey: Hosts and parasitoids in discrete generations, whose equilibrium is unstable to growing oscillations.
• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.
• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.
• Ricker map of overcompensating generations ricker: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.
• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.
• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.
• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.
• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.
• Wolf and sheep wolf_sheep: The wolf and sheep predation system with one grass patch per cell.

biology/epidemiology

• Activity-driven temporal networks activity_driven_epidemics: SIS spreading on an activity-driven temporal network, whose edges last one step.
• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Record-dynamics epidemic of resistant residents record_dynamics_epidemic: Yearly strains that infect every resident less resistant than their strength, who become immune.
• SIR sir: An epidemic that moves individuals from susceptible to infected to recovered.
• SIR on a network as an agent system network_sir: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.
• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.
• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS sis: An infection without immunity, where the recovered become susceptible again and the disease is endemic.
• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.
• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.
• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

biology/evolution

• Adaptive climbing in finite populations adaptive_climbing: A finite population that climbs a random fitness landscape by mutation and strong selection.
• Assembly growth along the assembly index assembly_growth: The numbers of objects at successive assembly indices, each discovered from the level below.
• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.
• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.
• Hypercycles hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.
• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.
• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.
• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.
• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Replicator dynamics and hawks and doves replicator_dynamics: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.
• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.
• Tangled Nature tangled_nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.
• Undirected and directed polymer assembly pool polymer_assembly_pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.

biology/neuroscience

• Adaptrode synapse adaptrode: An Adaptrode synapse driven by a train of input pulses that stops after a while.
• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.
• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.
• FitzHugh-Nagumo ODE fitzhugh_nagumo_ode: The FitzHugh-Nagumo neuron, the two-variable reduction of Hodgkin and Huxley's excitable membrane.
• Fukai-Tanaka winner-take-all network winners_take_all: A winner-take-all network of units with self-limited growth and mutual inhibition.
• Hodgkin-Huxley hodgkin_huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.
• Integrate-and-fire integrate_and_fire: The leaky integrate-and-fire neuron, tau v' = -(v - rest) + R I, which spikes and resets at a threshold.
• Izhikevich izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.
• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.
• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.
• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.
• Polyhomeostatic neuron polyhomeostatic_neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
• Theta neurons and QIF networks theta_neurons: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.
• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.
• Wilson-Cowan wilson_cowan: The Wilson-Cowan system of interacting excitatory and inhibitory populations of neurons.

biology/oncology

• Acquired and induced resistance with phenotype switching drug_resistance_switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.
• Bone-marrow and blood cells with blasts and cancer stem cells bone_marrow_blood_leukaemia: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.
• Cell-cycle compartments with killing, blocking, and recruiting agents cell_cycle_chemotherapy: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Exponential-linear tumour growth exponential_linear_growth: A tumour that grows exponentially while small and linearly once its growth reaches a limiting rate.
• Generalized logistic (Richards) growth generalized_logistic: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.
• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Gompertz growth gompertz: A tumour whose growth slows as the logarithm of the room left, dn/dt = r n log(K / n).
• Gompertz growth with a vascular carrying capacity dynamic_carrying_capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.
• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Immune competition immune_competition: The kinetic system of the competition between tumour cells and immune cells.
• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Kirschner-Panetta system with TGF-beta and siRNA sirna_tgf_beta: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Linear-quadratic cell survival and biologically effective dose lq_cell_survival: The linear-quadratic survival of cells after a total dose, with its biologically effective dose.
• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.
• Normal and tumour competition under radiation radiotherapy_competition: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.
• Normal, tumour, and immune cells under chemotherapy de_pillis_radunskaya: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Oncolytic adenovirus with T cells and dendritic-cell injection oncolytic_virus_dendritic: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Optimal chemotherapy dosing optimal_chemotherapy: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Power-law tumour growth power_law_growth: A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.
• Predator-prey tumour-immune system of TCR gene therapy tcr_gene_therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.
• Primary and secondary sites with a migration delay under chemotherapy delayed_metastasis: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.
• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
• Prostate cancer under continuous androgen suppression continuous_androgen_suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.
• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.
• Resistance subpopulations under three drugs multidrug_resistance: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.
• Resting and hunting immune cells with a conversion delay under chemotherapy delayed_resting_hunting_chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.
• Skipper-Schabel-Wilcox log-kill skipper_log_kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.
• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.
• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.
• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.
• Uninfected and infected tumour cells under an oncolytic virus oncolytic_virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.
• Von Bertalanffy growth von_bertalanffy: A tumour that grows by the balance of anabolism and catabolism, dn/dt = r n^p - d n.

biology/physiology

• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.
• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.
• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.
• Cumulative drug toxicity drug_toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• FitzHugh-Nagumo spirals fitzhugh_nagumo: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.
• Lumped cardiovascular system cardiovascular: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.
• Mackey-Glass mackey_glass: The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.
• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.
• Two-compartment pharmacokinetics two_compartment_pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.
• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.
• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.
• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

biology/systems_biology

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.
• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.
• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.
• Drosophila circadian clock circadian_clock: Goldbeter's system of the Drosophila circadian clock, PER that represses its own transcription.
• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.
• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.
• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.
• Flux balance, flux variability, and parsimonious FBA flux_balance: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.
• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.
• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.
• Genetic toggle switch toggle_switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Kauffman random Boolean networks random_boolean_network: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.
• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Negative autoregulation and bursty gene expression negative_autoregulation: A gene that represses its own transcription and makes its protein in bursts.
• Probabilistic Boolean network probabilistic_boolean_network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.
• Repressilator repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.
• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.
• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.
• Three-gene Boolean network three_gene_boolean_network: The three-gene Boolean network of Raman, updated synchronously.
• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.
• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.
• Yeast cell-cycle network yeast_cell_cycle: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.
• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.

chemistry/kinetics

• Bistability on a catalyst pellet catalyst_pellet: The surface concentration of a catalyst pellet, bistable through Langmuir-Hinshelwood kinetics.
• Brusselator ODE brusselator_ode: The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
• Chemical Langevin equation of the Schlögl reaction chemical_langevin_equation: The chemical Langevin equation of Schlögl's bistable reaction, whose noise switches between two states.
• Chemical Langevin equation of the enzyme network chemical_langevin_network: The chemical Langevin equation of the enzyme reaction E + S <-> C -> E + P.
• Cubic autocatalysis cubic_autocatalysis: Cubic autocatalysis A + 2B -> 3B with the decay B -> C in a continuously fed reactor.
• Cyclic cellular automaton cyclic_cellular_automaton: Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
• Hodgepodge machine hodgepodge: The hodgepodge machine, a system of a chemical reaction whose waves form spirals.
• Michaelis-Menten as ODEs and as a jump process michaelis_menten: The enzyme reaction E + S <-> C -> E + P of Michaelis and Menten, as rate equations or as jumps.
• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.
• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.
• Pulsed electrochemical reactor pulsed_electrochemical_reactor: A stirred electrochemical reactor whose electrode is driven by a train of current pulses.

chemistry/pattern_formation

• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Gray-Scott gray_scott: The Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.
• Schnakenberg schnakenberg: The Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.

computer_science/automata

• Adding machine (odometer) adding_machine: The adding machine r → r + 1 on the q-adic integers, the odometer acting on the q-ary tree.
• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.
• Automata on the Grigorchuk group grigorchuk_automaton: The majority automaton on a ball of the Grigorchuk group, a group of intermediate growth.
• Automata on the free group free_group_automaton: The majority automaton on a ball of the free group, a tree in which every cell follows its neighbours.
• Baumslag-Solitar groups baumslag_solitar_automaton: The majority automaton on a ball of the Baumslag-Solitar group BS(m, n), of exponential growth.
• Context-free subshift context_free_subshift: A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.
• Diffusion automaton diffusion_automaton: A cellular automaton of diffusion, c' = c + r Σ (c_neighbour - c), started from a square of ones.
• Elementary automata elementary_automaton: An elementary cellular automaton by its Wolfram number, started from a single live cell by default.
• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.
• Fredkin's self-replicating parity automaton fredkin_replicator: Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
• Game of Life game_of_life: Conway's Game of Life, or another life-like rule, on a torus.
• Game of Life on a rewired small-world network small_world_life: Conway's Game of Life on a small-world network rewired from the Moore lattice of a torus.
• Group-valued difference automaton group_difference_automaton: The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.
• Heisenberg heisenberg_automaton: The majority automaton on a ball of the discrete Heisenberg group, of polynomial growth of degree 4.
• Highlife and the planar automaton zoo highlife: HighLife, B36/S23, or any other life-like automaton of the planar zoo by its rule, on a torus.
• Involutive automaton involutive_automaton: A binary automaton on a ring that is its own inverse, with the minimal memory set {-1, 0, ..., n}.
• Lamplighter lamplighter_automaton: The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
• Langton's ant langtons_ant: Langton's ant, which wanders chaotically for about 10,000 steps before it builds a periodic highway.
• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.
• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.
• Linear automata over rings linear_ring_automaton: The discrete Laplacian as a linear automaton on a ring, over the field of modulus elements.
• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.
• Muller's pre-injective non-surjective automaton on F2 muller_automaton: Muller's automaton on the free group F2, pre-injective but not surjective.
• Nilpotent automaton of class n nilpotent_automaton: A binary automaton on a ring that is nilpotent of class n: n steps send every configuration to 0.
• Noisy automata noisy_automata: An elementary cellular automaton whose outputs are flipped at random with a small probability.
• Power-series automaton power_series_automaton: The automaton τ(x)(n) = x(n) - t x(n + 1) over the ring K[[t]] of formal power series on a ring.
• Real quadratic automaton real_quadratic_automaton: The automaton τ(x)(n) = x(n + 1) - x(n)^2 over the reals on a ring.
• Self-indexed shift over a group alphabet self_indexed_shift: The map τ(x)(g) = x(g · x(g)) over the group as alphabet, continuous and equivariant but no automaton.
• Subshifts subshifts: A Markov chain on the golden mean shift, the configurations of a ring that never hold two adjacent ones.
• Subtraction automaton over the naturals subtraction_automaton: The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.
• Turing machine (three-state busy beaver) turing_machine: A Turing machine, a head with an internal state that reads, writes, and moves on a tape of symbols.
• Unbounded-alphabet automaton unbounded_alphabet_automaton: An additive automaton over the nonnegative integers, x'(n) = x(n - 1) + x(n), from a single 1.

computer_science/evolutionary_computation

• Bit-string evolution on a fitness landscape bit_string_evolution: A population of bit strings that evolves on a random additive fitness landscape.
• RobbyGA can collector and evolution by circumstance robby_ga: Mitchell's genetic algorithm that evolves strategies for Robby, a robot that collects cans in a room.
• Simulated annealing, genetic algorithms, and differential evolution evolutionary_optimisers: A population of candidates that searches a two-dimensional landscape, one generation per step.

computer_science/neural_networks

• Asymmetric diluted Hopfield network diluted_hopfield: The asymmetric diluted Hopfield network, which recalls a stored pattern below its memory capacity.
• Binary perceptron and random K-SAT perceptron_satisfiability: A binary perceptron that must store random patterns, revealed one per step, as cardinality constraints.
• Echo state network echo_state_network: An echo state network, a reservoir of leaky tanh units with random recurrent weights driven by a sine.
• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.
• Hopfield network hopfield: A Hopfield network that stores patterns with Hebb's rule and recalls the first from a corrupted copy.
• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.

economics/econophysics

• Boltzmann wealth exchange boltzmann_wealth: A Boltzmann wealth exchange system where every agent starts with one unit.
• El Farol el_farol: Arthur's El Farol bar problem, whose attendance self-organizes around the capacity.
• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.
• Exchange with debt limit and with saving propensity kinetic_wealth_exchange: A kinetic wealth exchange system with a saving propensity or a debt limit.
• Kinetic wealth and welfare dynamics kinetic_wealth_welfare: The kinetic theory of wealth classes, with a welfare policy when the tax is positive.
• Kolkata Paise Restaurant kolkata_restaurant: The Kolkata Paise Restaurant problem, where diners spread over restaurants that serve one meal each.
• Minority game and its evolutionary version minority_game: The minority game of an odd number of agents with two random strategies each.

economics/finance

• Bouchaud-Mézard wealth bouchaud_mezard_wealth: Bouchaud and Mezard's wealth dynamics, random multiplicative returns with exchange between neighbours.
• Correlated geometric Brownian motions correlated_geometric_brownian_motion: Two assets whose prices follow geometric Brownian motions driven by correlated Wiener processes.
• DebtRank cascades debtrank: The DebtRank cascade of distress from a set of banks through their interbank loans.
• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.
• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.
• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.
• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Geometric Brownian motion geometric_brownian_motion: An asset whose price follows geometric Brownian motion.
• Gibrat gibrat_process: Firms that grow by Gibrat's law of proportionate effect, which makes their sizes lognormal.
• Kesten processes kesten_process: Firms whose sizes follow Kesten's process, a random multiplicative factor plus a random positive term.
• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.
• Multiplicative multiplicative_process: Levy and Solomon's multiplicative process, whose floor at a fraction of the mean makes a power-law tail.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.
• Random-walk traders with arbitrage arbitrage_traders: Schwichtenberg's diffusion of a good by random-walk traders, minimally coupled to exchange rates.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

economics/innovation

• Arthur-Polak technology evolution arthur_polak: Arthur and Polak's evolution of technology, which builds logic functions from NAND gates.
• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.
• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.
• Combinatorial creative phase transition creative_phase_transition: The diversity of a combinatorial system, which jumps from sparse to full above a critical rule density.
• Fixed-line phone adoption phone_adoption: Hamill and Gilbert's adoption of fixed-line phones by British households from 1951.
• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• NK technology system nk_technology: McNerney's system of technological improvement, whose cost falls as a power of the cumulative production.
• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.

economics/macroeconomics

• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.
• Capital stock capital_stock: Meadows' capital stock, grown by reinvested output and drained by depreciation.
• Exchange rates in international trade exchange_rates: Hamill and Gilbert's staged adjustment of trade and the exchange rate to a rise in home prices.
• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.
• Fractional-reserve banking fractional_reserve_banking: Customers and banks whose lending of excess reserves multiplies the base money by 1 / ratio.
• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.
• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.
• Multiplier-accelerator multiplier_accelerator: Samuelson's multiplier-accelerator, Y_{t+1} = c (1 + v) Y_t - c v Y_{t-1} + G.
• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.
• Solow solow: An economy whose capital per worker approaches its steady state through saving and depreciation.
• Stock-flow consistent monetary economy stock_flow_consistent_economy: A stock-flow consistent economy, a closed circuit of money between households, firms, and a bank.
• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.
• Toy firm investment toy_firm_investment: Heterogeneous firms that invest out of past profits and sell at random prices, with entry and exit.

economics/markets

• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.
• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.
• Brock-Hommes heuristic switching brock_hommes: Brock and Hommes' asset market with traders switching between fundamentalists and trend chasers.
• Car wash queue car_wash: Chesney's Clean and Wax hand car wash, a shadow account of the revenue of its workers' labour.
• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Cobweb cobweb: A market whose price converges to or oscillates around equilibrium as supply lags the price.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Cournot duopoly with beliefs cournot_duopoly: Two Cournot duopolists that adjust their output to their beliefs about each other.
• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.
• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.
• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.
• Fair-trade food choice with peer influence fair_food: The choice between fair-trade and regular food of consumers who influence their neighbours.
• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.
• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.
• Labour market labour_market: Hamill and Gilbert's labour market of firms posting wage offers and unemployed workers searching.
• Produce market produce_market: Hamill and Gilbert's produce market, where shoppers with lists of produce visit stalls at random.
• Riccati price adjustment riccati_prices: A price that moves with excess demand under linear demand and saturating quadratic supply.
• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.
• Static oligopoly equilibria oligopoly_equilibria: The equilibria of a Cournot duopoly and of a pricing game of firms with loyal customers and switchers.
• Sugarscape sugarscape: The Sugarscape of Epstein and Axtell, foragers of random vision and metabolism on two sugar peaks.
• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.
• Zero-intelligence double auction double_auction: Gode and Sunder's double auction of zero-intelligence traders, a step per trading period.

economics/resources

• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.
• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.
• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.
• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Harvesting farmers harvesting: A socio-ecological system of farmer agents harvesting a logistically regrowing resource.
• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.
• Oil economy oil_economy: Meadows' oil economy, capital that grows on a nonrenewable resource and collapses as it runs out.
• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.
• Renewable fishery with capital fishery_capital: Meadows' fishing fleet, capital that grows on a renewable resource and can overshoot it.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.
• Von Thünen land use von_thunen: A square plain around one market at its centre, whose every cell grows the crop of highest bid rent.

engineering/circuits

• Chua double and multiple scrolls chua: Chua's circuit, whose double-scroll or multiscroll attractor is chaotic.
• Circuits under periodic impulses impulsive_circuits: An RLC circuit driven by a periodic train of current impulses, L C x'' + R C x' + x = A Σ δ(t - k T).
• Coupled Colpitts oscillators coupled_colpitts: Two detuned chaotic Colpitts oscillators coupled through the difference of their second variables.
• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Memristor oscillator memristor_oscillator: The simplest memristor oscillator: a capacitor, an inductor, and a memristor in series.
• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.
• RLC circuit rlc_circuit: A series RLC circuit written as Kirchhoff's voltage law, L q'' + R q' + q / C = V cos(ω t).
• Tunnel-diode oscillator tunnel_diode_oscillator: A tunnel-diode oscillator: an inductor and a capacitor in parallel with a negative-resistance diode.
• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.

engineering/control

• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.
• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.
• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.
• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.
• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.
• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.
• Satellite transfer satellite_transfer: A satellite moved between Kepler orbits of equal angular momentum by radial thrust.
• Thermostat thermostat: A room heated by a furnace under a thermostat while heat leaks to the outside.

engineering/mechatronics

• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.
• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.
• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.
• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.

engineering/power_systems

• Faraday-disk charger faraday_disk: A battery charger driven by a Faraday disk generator whose own current strengthens its field.
• Regenerative braking regenerative_braking: A car braked by a Faraday disk generator on its wheel, which charges a supercapacitor.
• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.
• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.

engineering/processes

• Bathtub bathtub: A bathtub, one stock of water with an inflow from the faucet and an outflow through the drain.
• Cascade classifier cascade_classifier: A cascade classifier, a column of stages that separates a narrow size class of particles by an air flow.
• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.
• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.
• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.

engineering/robotics

• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Light-sensitive robots with sensory delay light_seeking_robots: Light-sensitive robots in a circular well, whose sensory delay decides whether they gather in the light.
• Swarmtaxis swarmtaxis: The swarmtaxis algorithm, which moves a swarm of robots towards a beacon that only some of them see.

engineering/traffic

• Car following with delay delayed_car_following: A chain of cars that react to the speed difference to the car ahead after a reaction time.
• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Intelligent driver system intelligent_driver_system: The intelligent driver system of cars on a ring road.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Langevin car-following toward a density-dependent speed langevin_traffic: Cars on a ring road whose speeds relax with noise towards a desired speed set by the local density.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Nagel-Schreckenberg nagel_schreckenberg: The Nagel-Schreckenberg traffic system on a ring road, where jams emerge above a critical density.
• Optimal velocity optimal_velocity: The optimal velocity system of cars on a ring road, where uniform flow breaks into stop-and-go waves.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.
• Ring-road driver agents ring_road_drivers: Drivers on a circular highway, where small random errors of speed set off traffic jams.

mathematics/dynamical_systems

• Baker baker_map: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
• Bernoulli doubling map doubling_map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.
• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
• Cat map cat_map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.
• Circle circle_map: The sine circle map θ' = θ + Ω - K / (2π) sin 2πθ (mod 1), which locks into Arnold tongues.
• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
• Cubic chaotic map and its chaotic walk cubic_chaotic_map: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• Fermi-Ulam map fermi_ulam_map: The Fermi-Ulam map of a ball bouncing between a fixed wall and a wall that oscillates in time.
• Finite-time blow-up finite_time_blow_up: Hyperbolic growth x' = a x^p, which diverges at the finite time x0^(1 - p) / (a (p - 1)) for p > 1.
• Generalized tent map skew_tent_map: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.
• Hopf hopf_form: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
• Hyperchaotic four-dimensional Rössler flow hyperchaotic_rossler: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
• Hénon map henon_map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
• Ikeda delay equation ikeda: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
• Intermittency maps intermittency_map: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
• Langford attractor langford: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
• Logistic map logistic_map: The logistic map x_{t+1} = r x_t (1 - x_t), which reaches chaos through period doubling as r grows.
• Lorenz lorenz: The Lorenz-63 system of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
• Lozi map lozi_map: The Lozi map x' = 1 + y - a |x|, y' = b x, the piecewise linear Hénon map with a strange attractor.
• Newton-Raphson map newton_raphson_map: Newton's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.
• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.
• Quadratic system with a slow manifold and a finite Koopman embedding koopman_slow_manifold: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.
• Rössler rossler: The Rössler system, a chaotic flow with one folded band for its standard parameters.
• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
• Saddle-node saddle_node_form: The saddle-node normal form x' = μ - x², whose two equilibria ±√μ meet and vanish at μ = 0.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.
• Sprott jerk sprott_jerk: Sprott's jerk equation x''' = -A x'' - x' + |x| - 1, chaotic near A = 0.6.
• Standard standard_map: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Takens-Bogdanov takens_bogdanov_form: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.
• Tent tent_map: The tent map x' = μ min(x, 1 - x), chaotic with Lyapunov exponent ln μ for μ above 1.
• Thomas cyclically symmetric attractor thomas: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.
• Towel maps towel_map: Rössler's towel map, a three-dimensional map with two positive Lyapunov exponents.

mathematics/fractals

• Cantor cantor_set: The Cantor set as the substitution of every interval by its two outer parts.
• Iterated function systems by the chaos game chaos_game: Points that draw the attractor of an iterated function system by the chaos game.
• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• Koch koch_curve: The Koch curve as the L-system F → F+F--F+F drawn by a turtle.
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).
• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.
• Sierpiński substitutions sierpinski_substitution: The Sierpiński triangle or carpet as the substitution of a polygon by scaled copies of itself.

mathematics/fractional_dynamics

• Fractional Lorenz fractional_lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
• Fractional Van der Pol fractional_van_der_pol: The fractional Van der Pol oscillator D^alpha x = y, D^alpha y = mu (1 - x^2) y - x.
• Fractional damped oscillator fractional_oscillator: The fractional damped oscillator D^alpha x = v, D^alpha v = -omega^2 x - 2 gamma v.
• Fractional relaxation fractional_relaxation: Fractional relaxation D^alpha x = -lambda x, which decays as the Mittag-Leffler function E_alpha.

mathematics/networks

• Barabási-Albert growth, with Bianconi-Barabási fitness barabasi_albert: Barabási-Albert growth, whose degrees follow a power law P(k) ~ k^-3 by preferential attachment.
• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.
• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.
• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.
• Diffusion network_diffusion: Diffusion on a network, du_i/dt = D Σ_j A_ij (u_j - u_i), which spreads a unit mass to every node.
• Erdos-Renyi and Watts-Strogatz graphs random_graph_ensembles: The construction of an Erdős-Rényi or a Watts-Strogatz random graph, one step of new edges at a time.
• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.
• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.
• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.
• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.
• Kronecker graphs kronecker_graph: The Kronecker graph of two graphs, whose adjacency matrix is the Kronecker product of theirs.
• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.
• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.
• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.
• PageRank iterations on networks pagerank: The PageRank iteration r_i' = (1 - d) / n + d Σ_j A_ij r_j / k_j on an undirected network.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.
• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.
• Random walks network_random_walk: Random walkers on a network, which visit every node of a connected graph in proportion to its degree.
• Size-biased random walk on hypergraphs hypergraph_random_walk: The size-biased random walk on a hypergraph, p_j' = sum_i p_i T_ij - p_j, from one node.
• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.
• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.
• Vertex-copying network growth vertex_copying: Vertex-copying growth, whose new nodes copy the edges of existing nodes and make a heavy-tailed degree.

mathematics/stochastic_processes

• Ageing random walk ageing_random_walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.
• Bernoulli process bernoulli_process: Independent Bernoulli processes, one success or failure per step, whose success counts are binomial.
• Birth-death process and M/M/1 queue birth_death_queue: Independent M/M/1 queues, birth-death processes with constant arrival and service rates.
• Cascading sample-space-reducing process cascading_sample_space_reducing: Cascading sample-space-reducing processes, whose visits follow the power law i^(-mu) for any mu.
• Exponential growth observed at exponential random times exponential_growth_at_random_times: Exponential growth observed at exponentially distributed times, whose observed values follow a power law.
• Fragmentation fragmentation: The random fragmentation of an object of integer size, whose fragments keep breaking into smaller ones.
• Galton-Watson branching galton_watson: A Galton-Watson branching process with Poisson offspring, one generation per step.
• Hidden Markov source hidden_markov_source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.
• Markov source markov_source: A Markov source, a sequence of symbols in which each symbol depends on the one before it.
• Ornstein-Uhlenbeck with weakening restoring force weakening_ornstein_uhlenbeck: Brownian particles held by a restoring force that weakens with distance.
• Persistent random walk persistent_random_walk: Walkers on the integers that keep the direction of their last step with a probability.
• Poisson process poisson_process: Independent Poisson processes, counters that grow by one at exponentially distributed waiting times.
• Pólya urn polya_urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.
• Random two-branch interval map random_branch_map: Orbits of a random map of [0, 1] with the two branches (1 + x) / 2 and mu x².
• Random walk with autoregressive velocity random_walk_with_memory: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.
• Random walks random_walk: Independent walkers on the integers that step at random times, a continuous-time random walk.
• Sample-space-reducing process sample_space_reducing: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.

physics/active_matter

• Active Brownian particles active_brownian_particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Active Ornstein-Uhlenbeck particles active_ornstein_uhlenbeck_particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Chiral active particles chiral_active_particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.
• Cucker-Smale flocking cucker_smale_flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.
• Depot-pumped Brownian particles depot_brownian_particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.
• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.
• Motility-induced phase separation motility_induced_phase_separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.
• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Run-and-tumble run_and_tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.
• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.
• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
• Toner-Tu hydrodynamics toner_tu: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

physics/celestial_mechanics

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• N-body and solar system n_body: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Two-body orbit two_body_orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.

physics/climate

• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Coupled soil and deep temperature maps soil_temperature: The coupled maps of the soil surface temperature and the deeper soil temperature.
• Daisyworld daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.
• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• ENSO delay oscillator enso: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).
• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Gissinger geodynamo geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.
• Hasselmann stochastic climate hasselmann: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.
• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Lorenz-84 lorenz_84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.
• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Rothman excitable carbon cycle carbon_cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.
• Saltzman-Maasch glaciation saltzman_glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.
• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.
• Zero-dimensional energy balance with ice albedo energy_balance: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.

physics/cosmology

• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.

physics/criticality

• Bak-Tang-Wiesenfeld sandpile sandpile: The Bak-Tang-Wiesenfeld sandpile on a lattice with open edges, started from random stable heights.
• Drossel-Schwabl forest fire forest_fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.
• Manna sandpile manna_sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

physics/fluids

• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Decaying turbulence decaying_turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.
• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• FHP lattice gas fhp_lattice_gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.
• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

physics/heat

• Cooling coffee cooling_coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).
• Heat diffusion heat_diffusion: Heat diffusing on a plate from a hot square in its middle, du/dt = D laplacian(u).
• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.

physics/mechanics

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Bouncing ball bouncing_ball: A ball that falls under gravity and bounces off the floor, losing speed by a coefficient of restitution.
• Circular billiard circular_billiard: A particle moving freely inside a circle and reflecting specularly off its wall.
• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Harmonic oscillator harmonic_oscillator: A particle on a spring, optionally damped.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Magnetic pendulum magnetic_pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• Newton's cradle newtons_cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.
• Particle in the well k x^2 e^-x particle_in_a_well: A particle moving without friction in the potential U(x) = k x² e^-x.
• Pendulum pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Projectile projectile: A particle launched under uniform gravity, optionally slowed by linear drag.
• Rotating balls on rods rotating_balls_on_rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.

physics/nonlinear_oscillators

• Adler equation adler_equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Duffing duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• Dynamical billiards of Sinai, mushroom, and open three-disk shapes billiards: A particle moving freely on a billiard table and reflecting specularly off its walls.
• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Hénon-Heiles henon_heiles: The Hénon-Heiles system of a star in a galactic potential, whose motion is chaotic at higher energies.
• Nuclear quadrupole vibrations nuclear_quadrupole: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.
• Stuart-Landau stuart_landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.
• Van der Pol van_der_pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.

physics/oceanography

• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.
• Stommel thermohaline boxes stommel: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.

physics/pattern_formation

• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.
• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.

physics/seismology

• Burridge-Knopoff spring-block fault burridge_knopoff: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Olami-Feder-Christensen earthquakes: The Olami-Feder-Christensen earthquake system with open edges, started from random forces.

physics/soft_matter

• Annealed transient time motion annealed_transient_time_motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Brownian particle near a wall brownian_near_wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Continuous-time random walk continuous_time_random_walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Diffusion in a speckle field diffusion_in_a_speckle_field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.
• Fractional Brownian motion fractional_brownian_motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.
• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.
• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.
• Inertial and overdamped Brownian motion brownian_motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.
• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.
• Langevin equation with colored noise colored_noise_langevin: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.
• Lévy walk levy_walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).
• Multiplicative noise conventions multiplicative_noise: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.
• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Optical trap including oscillating and blinking traps optical_trap: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.
• Overdamped particle in a tilted periodic potential tilted_periodic_potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.
• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.
• Scaled Brownian motion scaled_brownian_motion: Scaled Brownian motion, dx = sqrt(2 K alpha (t + t0)^(alpha - 1)) dW, with <x^2> ~ 2 K t^alpha.
• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.
• Stochastic resonance stochastic_resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.
• Superdiffusive Levy flight levy_flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

physics/statistical_mechanics

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Bouchaud trap system trap_system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.
• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Continuum percolation continuum_percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Diffusion-limited aggregation diffusion_limited_aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Discrete master equation discrete_master_equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.
• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Ehrenfest urn system of two boxes ehrenfest_urn: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.
• Fully connected Ising system with Glauber dynamics curie_weiss: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.
• Ideal gas ideal_gas: An ideal gas of non-interacting molecules in a reflecting box.
• Ising ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Ising on a graph graph_ising: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Landau mean-field relaxation landau_relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Lennard-Jones fluid lennard_jones_fluid: Lennard-Jones particles in a periodic square box, optionally held at a temperature by a thermostat.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Potts potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• Q2R reversible Ising automaton q2r: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.
• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Random energy system random_energy_system: The random energy system, 2^N quenched Gaussian energies of variance N / 2 at a temperature.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.
• Site percolation site_percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.
• Two-phase zone exchange two_phase_zone_exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.
• XY xy: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.
• Zero-range process zero_range_process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

physics/synchronization

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.
• Coupled map lattices on networks coupled_map_lattices_on_networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Delayed Kuramoto delayed_kuramoto: Kuramoto phase oscillators coupled all to all through signals that arrive after a delay.
• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Kuramoto kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Kuramoto on networks kuramoto_on_networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Kuramoto oscillators with three-body coupling hypergraph_kuramoto: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Kuramoto-Sakaguchi chimera kuramoto_sakaguchi_chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• Mirollo-Strogatz pulse-coupled oscillators mirollo_strogatz: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.
• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.
• Ott-Antonsen reduction ott_antonsen: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.
• Rössler-driven Lorenz and generalized synchronization rossler_driven_lorenz: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.
• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.
• Winfree system winfree: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

physics/waves

• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.
• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.
• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Standing waves on a string standing_waves: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.

social/cooperation

• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.
• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.
• Iterated prisoner's dilemma tournament prisoners_dilemma_tournament: The evolution of strategies of the iterated prisoner's dilemma from random genomes.
• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Public goods game public_goods_game: The spatial public goods game in groups of five neighbours, with Fermi imitation.
• Public goods game on hypergraphs hypergraph_public_goods: The public goods game on a hypergraph, played in every hyperedge, with Fermi imitation.
• Spatial prisoner's dilemma with best-neighbour and Fermi imitation spatial_prisoners_dilemma: The spatial prisoner's dilemma of Nowak and May, with best-neighbour or Fermi imitation.
• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.
• Two-player games on links and three-body games on 2-simplices simplicial_social_dilemmas: The social dilemmas of Guo et al. on the links and the 2-simplices of a simplicial complex.

social/crowds

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Criminality and security criminality: The kinetic system of criminality in a society and the security forces that act against it.
• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.
• Social force pedestrians social_force: Two crowds walking in opposite directions along a periodic corridor, which form lanes.

social/language

• Mandelbrot least-effort word frequencies mandelbrot_least_effort: The word frequencies that minimise the mean cost of a language over its entropy, a power law of rank.
• Miller's random typewriting random_typewriting: Miller's random typewriting, monkeys whose random keystrokes give Zipf's law of word frequencies.

social/opinion_dynamics

• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.
• Belief diffusion with influentials influentials: The belief diffusion system with influentials who care passionately and weigh more with their neighbours.
• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.
• Cognitive consistency agents cognitive_consistency: Agents whose attitudes spread by conversation and settle in a belief network that seeks consistency.
• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
• Deffuant bounded confidence deffuant: The Deffuant bounded confidence system, whose opinions settle into about 1 / (2 * confidence) clusters.
• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.
• Galam majority galam_majority: The Galam majority system, where random groups adopt their majority until one opinion wins.
• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.
• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.
• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
• Kinetic opinion formation kinetic_opinion: The kinetic system of opinion formation over a discrete grid of opinions.
• Majority rule automaton majority_rule: Voters that take the majority opinion of their neighbourhood, which coarsens into domains.
• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.
• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Naming game naming_game: The minimal naming game, where agents with empty inventories agree on one word.
• Nonlinear three-body consensus on temporal hypergraphs hypergraph_consensus: The nonlinear three-body consensus system on a random hypergraph of triangles that switch on and off.
• Social impact theory social_impact: The social impact system on a full grid, with random persuasiveness and supportiveness.
• Sznajd sznajd: The Sznajd system, where agreeing pairs of agents convince their neighbours.
• Three-state opinion with undecided agents three_state_opinion: The three-state opinion system of agents against, undecided, and for, on a random regular network.
• Threshold adoption threshold_adoption: The threshold adoption system in social circles, started by the 2.5% of agents with the lowest thresholds.
• Voter system voter_system: A voter system with one voter per node of a network.
• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.

social/organizations

• Ajani's Choice ajanis_choice: Chesney's Ajani's Choice, migrant farm workers and employers among whom underpayment emerges.
• Barabási priority queues priority_queues: The Barabási priority queue, whose waiting times follow a power law as the choice becomes greedy.
• Bonabeau hierarchies bonabeau_hierarchies: The Bonabeau system, where a hierarchy of strengths emerges above a critical density of agents.
• Brothel movement system trafficking: The brothel movement system, where victims moved between brothels leave traces in bills and train trips.
• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.
• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.
• Employment takeover employment_takeover: The employment takeover system on a periodic checkerboard, with half of each kind coloured at random.
• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.
• Exploration and preferential return exploration_preferential_return: The exploration and preferential return model of human mobility of Song and coauthors.
• Gravity interaction system of urban scaling gravity_urban_scaling: The residents of a city at uniform positions in a disc, each interacting with the others as r^-gamma.
• Hora and Tempus watchmakers watchmakers: Simon's fable of the watchmakers Hora and Tempus, told by Meadows, with and without stable subassemblies.
• Insider sabotage insider_sabotage: The insider sabotage system of employees whose risk of sabotage builds up over the days of a year.
• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.
• Obesity system obesity: The obesity system of adults who eat near home and work, one step per day.
• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.
• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.
• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.

social/segregation

• Schelling schelling: A Schelling segregation system on a grid with at most one resident per cell.
• Stochastic Schelling stochastic_schelling: The stochastic Schelling system of agents moving between blocks, which segregates above beta = 2.

social/spreading

• Maki-Thompson rumours maki_thompson: The Maki-Thompson rumour system started by one spreader, which leaves about 20% of people ignorant.
• SIS on hypergraphs with heterogeneous exposure and a nonlinear infection kernel hypergraph_exposure_sis: The SIS contagion of St-Onge et al. in the environments of a hypergraph, with a nonlinear kernel.
• Simon-Yule growth simon_yule: The Simon system of a text that grows from one word, whose word counts follow a power law.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.
• Watts threshold cascades watts_cascades: The Watts threshold system of global cascades started by a few random seeds.

By tag

Achlioptas process

• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.

B-tipping

• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.

Belousov-Zhabotinsky

• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.

COMETS

• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.

ENSO

• ENSO delay oscillator enso: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).

Edgeworth box

• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.

El Niño

• ENSO delay oscillator enso: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).

Gaia

• Daisyworld daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.

Garden of Eden

• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.
• Muller's pre-injective non-surjective automaton on F2 muller_automaton: Muller's automaton on the free group F2, pre-injective but not surjective.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.

Gompertz

• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.

HER2

• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.

Hill function

• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.

IL-2

• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.

Koopman operator

• Quadratic system with a slow manifold and a finite Koopman embedding koopman_slow_manifold: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.

L-system

• Koch koch_curve: The Koch curve as the L-system F → F+F--F+F drawn by a turtle.

Langmuir-Hinshelwood

• Bistability on a catalyst pellet catalyst_pellet: The surface concentration of a catalyst pellet, bistable through Langmuir-Hinshelwood kinetics.

Laplacian

• Linear automata over rings linear_ring_automaton: The discrete Laplacian as a linear automaton on a ring, over the field of modulus elements.

Lotka-Volterra

• Generalized Lotka-Volterra microbial community generalized_lotka_volterra: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).
• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.

Markov chain

• Subshifts subshifts: A Markov chain on the golden mean shift, the configurations of a ring that never hold two adjacent ones.

Michaelis-Menten

• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.

Monte Carlo

• Context-free subshift context_free_subshift: A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.
• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Ising ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Ising on a graph graph_ising: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Potts potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• XY xy: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.

N-tipping

• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.

NK system

• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.

Ornstein-Uhlenbeck

• Hasselmann stochastic climate hasselmann: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.
• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.

PD-1

• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.

PSA

• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.

Perron eigenvalue

• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.

R-tipping

• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.

Richards

• Generalized logistic (Richards) growth generalized_logistic: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.

Rössler

• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.

S-gap

• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.

S-system

• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.

SteadyCom

• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.

Thue-Morse

• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.

Turing

• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Schnakenberg schnakenberg: The Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.

Vroman effect

• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.

absorbing state

• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.
• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.

absorbing states

• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.

acausal components

• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.
• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.

accelerator

• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.

active gel

• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.

active matter

• Active Brownian particles active_brownian_particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Active Ornstein-Uhlenbeck particles active_ornstein_uhlenbeck_particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.
• Bird swarm with pair potentials bird_swarm: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Chiral active particles chiral_active_particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.
• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.
• Depot-pumped Brownian particles depot_brownian_particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Light-sensitive robots with sensory delay light_seeking_robots: Light-sensitive robots in a circular well, whose sensory delay decides whether they gather in the light.
• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.
• Motility-induced phase separation motility_induced_phase_separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.
• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.
• Run-and-tumble run_and_tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.
• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.
• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

active turbulence

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.

activity-driven

• Activity-driven temporal networks activity_driven_epidemics: SIS spreading on an activity-driven temporal network, whose edges last one step.

adaptation

• Polyhomeostatic neuron polyhomeostatic_neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.
• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.

adaptive delays

• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.

adaptive network

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.
• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.
• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.
• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.
• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.
• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.

adaptive quotes

• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.

addiction

• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.

additive

• Unbounded-alphabet automaton unbounded_alphabet_automaton: An additive automaton over the nonnegative integers, x'(n) = x(n - 1) + x(n), from a single 1.

adjacent possible

• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.

adoption

• Fixed-line phone adoption phone_adoption: Hamill and Gilbert's adoption of fixed-line phones by British households from 1951.

adoptive therapy

• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.

age structure

• Fibonacci rabbits fibonacci_rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.

ageing

• Ageing random walk ageing_random_walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.

agents

• Active walkers and trail formation active_walkers: Active walkers who walk between the corners of a green and wear a system of trails into it.
• Adaptive climbing in finite populations adaptive_climbing: A finite population that climbs a random fitness landscape by mutation and strong selection.
• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.
• Ageing random walk ageing_random_walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.
• Agent-based Lotka-Volterra agent_based_lotka_volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Ajani's Choice ajanis_choice: Chesney's Ajani's Choice, migrant farm workers and employers among whom underpayment emerges.
• Ant colony optimization on a road network ant_colony_optimization: Ants that find the shortest path between two points of a Delaunay road network, round after round.
• Arthur-Polak technology evolution arthur_polak: Arthur and Polak's evolution of technology, which builds logic functions from NAND gates.
• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.
• Barabási priority queues priority_queues: The Barabási priority queue, whose waiting times follow a power law as the choice becomes greedy.
• Belief diffusion with influentials influentials: The belief diffusion system with influentials who care passionately and weigh more with their neighbours.
• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Bernoulli process bernoulli_process: Independent Bernoulli processes, one success or failure per step, whose success counts are binomial.
• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.
• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.
• Bit-string evolution on a fitness landscape bit_string_evolution: A population of bit strings that evolves on a random additive fitness landscape.
• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions boids: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Boltzmann wealth exchange boltzmann_wealth: A Boltzmann wealth exchange system where every agent starts with one unit.
• Bonabeau hierarchies bonabeau_hierarchies: The Bonabeau system, where a hierarchy of strengths emerges above a critical density of agents.
• Brock-Hommes heuristic switching brock_hommes: Brock and Hommes' asset market with traders switching between fundamentalists and trend chasers.
• Brothel movement system trafficking: The brothel movement system, where victims moved between brothels leave traces in bills and train trips.
• Car wash queue car_wash: Chesney's Clean and Wax hand car wash, a shadow account of the revenue of its workers' labour.
• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.
• Cascading sample-space-reducing process cascading_sample_space_reducing: Cascading sample-space-reducing processes, whose visits follow the power law i^(-mu) for any mu.
• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.
• Cognitive consistency agents cognitive_consistency: Agents whose attitudes spread by conversation and settle in a belief network that seeks consistency.
• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.
• Cournot duopoly with beliefs cournot_duopoly: Two Cournot duopolists that adjust their output to their beliefs about each other.
• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.
• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.
• DebtRank cascades debtrank: The DebtRank cascade of distress from a set of banks through their interbank loans.
• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.
• Deffuant bounded confidence deffuant: The Deffuant bounded confidence system, whose opinions settle into about 1 / (2 * confidence) clusters.
• Diffusion-limited aggregation diffusion_limited_aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.
• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.
• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.
• El Farol el_farol: Arthur's El Farol bar problem, whose attendance self-organizes around the capacity.
• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.
• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.
• Exchange with debt limit and with saving propensity kinetic_wealth_exchange: A kinetic wealth exchange system with a saving propensity or a debt limit.
• Exploration and preferential return exploration_preferential_return: The exploration and preferential return model of human mobility of Song and coauthors.
• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.
• Fair-trade food choice with peer influence fair_food: The choice between fair-trade and regular food of consumers who influence their neighbours.
• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.
• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.
• Fixed-line phone adoption phone_adoption: Hamill and Gilbert's adoption of fixed-line phones by British households from 1951.
• Fractional-reserve banking fractional_reserve_banking: Customers and banks whose lending of excess reserves multiplies the base money by 1 / ratio.
• Fragmentation fragmentation: The random fragmentation of an object of integer size, whose fragments keep breaking into smaller ones.
• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Galam majority galam_majority: The Galam majority system, where random groups adopt their majority until one opinion wins.
• Galton-Watson branching galton_watson: A Galton-Watson branching process with Poisson offspring, one generation per step.
• Gibrat gibrat_process: Firms that grow by Gibrat's law of proportionate effect, which makes their sizes lognormal.
• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.
• Harvesting farmers harvesting: A socio-ecological system of farmer agents harvesting a logistically regrowing resource.
• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.
• Hopfield network hopfield: A Hopfield network that stores patterns with Hebb's rule and recalls the first from a corrupted copy.
• Hora and Tempus watchmakers watchmakers: Simon's fable of the watchmakers Hora and Tempus, told by Meadows, with and without stable subassemblies.
• Insider sabotage insider_sabotage: The insider sabotage system of employees whose risk of sabotage builds up over the days of a year.
• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• Iterated prisoner's dilemma tournament prisoners_dilemma_tournament: The evolution of strategies of the iterated prisoner's dilemma from random genomes.
• Kesten processes kesten_process: Firms whose sizes follow Kesten's process, a random multiplicative factor plus a random positive term.
• Kolkata Paise Restaurant kolkata_restaurant: The Kolkata Paise Restaurant problem, where diners spread over restaurants that serve one meal each.
• Labour market labour_market: Hamill and Gilbert's labour market of firms posting wage offers and unemployed workers searching.
• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.
• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.
• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.
• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.
• Miller's random typewriting random_typewriting: Miller's random typewriting, monkeys whose random keystrokes give Zipf's law of word frequencies.
• Minority game and its evolutionary version minority_game: The minority game of an odd number of agents with two random strategies each.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.
• Multiplicative multiplicative_process: Levy and Solomon's multiplicative process, whose floor at a fraction of the mean makes a power-law tail.
• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.
• NK technology system nk_technology: McNerney's system of technological improvement, whose cost falls as a power of the cumulative production.
• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Naming game naming_game: The minimal naming game, where agents with empty inventories agree on one word.
• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.
• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.
• Obesity system obesity: The obesity system of adults who eat near home and work, one step per day.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.
• Pheromone foraging pheromone_foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Produce market produce_market: Hamill and Gilbert's produce market, where shoppers with lists of produce visit stalls at random.
• Random walks network_random_walk: Random walkers on a network, which visit every node of a connected graph in proportion to its degree.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.
• Ring-road driver agents ring_road_drivers: Drivers on a circular highway, where small random errors of speed set off traffic jams.
• RobbyGA can collector and evolution by circumstance robby_ga: Mitchell's genetic algorithm that evolves strategies for Robby, a robot that collects cans in a room.
• Run-and-tumble run_and_tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.
• SIR on a network as an agent system network_sir: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.
• Sample-space-reducing process sample_space_reducing: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.
• Schelling schelling: A Schelling segregation system on a grid with at most one resident per cell.
• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.
• Simon-Yule growth simon_yule: The Simon system of a text that grows from one word, whose word counts follow a power law.
• Social impact theory social_impact: The social impact system on a full grid, with random persuasiveness and supportiveness.
• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.
• Stochastic Schelling stochastic_schelling: The stochastic Schelling system of agents moving between blocks, which segregates above beta = 2.
• Strömbom shepherding of a flock shepherding: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.
• Sugarscape sugarscape: The Sugarscape of Epstein and Axtell, foragers of random vision and metabolism on two sugar peaks.
• Swarmtaxis swarmtaxis: The swarmtaxis algorithm, which moves a swarm of robots towards a beacon that only some of them see.
• Sznajd sznajd: The Sznajd system, where agreeing pairs of agents convince their neighbours.
• Tangled Nature tangled_nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.
• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.
• Three-state opinion with undecided agents three_state_opinion: The three-state opinion system of agents against, undecided, and for, on a random regular network.
• Threshold adoption threshold_adoption: The threshold adoption system in social circles, started by the 2.5% of agents with the lowest thresholds.
• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.
• Toy firm investment toy_firm_investment: Heterogeneous firms that invest out of past profits and sell at random prices, with entry and exit.
• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.
• Undirected and directed polymer assembly pool polymer_assembly_pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.
• Voter system voter_system: A voter system with one voter per node of a network.
• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.
• Watts threshold cascades watts_cascades: The Watts threshold system of global cascades started by a few random seeds.
• Wolf and sheep wolf_sheep: The wolf and sheep predation system with one grass patch per cell.
• Zero-intelligence double auction double_auction: Gode and Sunder's double auction of zero-intelligence traders, a step per trading period.

aggregation

• Diffusion-limited aggregation diffusion_limited_aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.
• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.

aging

• Bouchaud trap system trap_system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.

agriculture

• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.
• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.
• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.

alignment

• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Cucker-Smale flocking cucker_smale_flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

allometry

• Gravity interaction system of urban scaling gravity_urban_scaling: The residents of a city at uniform positions in a disc, each interacting with the others as r^-gamma.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.
• Von Bertalanffy growth von_bertalanffy: A tumour that grows by the balance of anabolism and catabolism, dn/dt = r n^p - d n.
• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

amplification

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.

amplitude death

• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.

androgen

• Prostate cancer under continuous androgen suppression continuous_androgen_suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.

androgen receptor

• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.

angiogenesis

• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Gompertz growth with a vascular carrying capacity dynamic_carrying_capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.
• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.

angular momentum

• Rotating balls on rods rotating_balls_on_rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.

anisotropy

• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.

annealing

• Simulated annealing, genetic algorithms, and differential evolution evolutionary_optimisers: A population of candidates that searches a two-dimensional landscape, one generation per step.

anomalous diffusion

• Annealed transient time motion annealed_transient_time_motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Continuous-time random walk continuous_time_random_walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Diffusion in a speckle field diffusion_in_a_speckle_field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.
• Fractional Brownian motion fractional_brownian_motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.
• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.
• Lévy walk levy_walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).
• Scaled Brownian motion scaled_brownian_motion: Scaled Brownian motion, dx = sqrt(2 K alpha (t + t0)^(alpha - 1)) dW, with <x^2> ~ 2 K t^alpha.
• Superdiffusive Levy flight levy_flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

anomalous relaxation

• Fractional relaxation fractional_relaxation: Fractional relaxation D^alpha x = -lambda x, which decays as the Mittag-Leffler function E_alpha.

ant

• Langton's ant langtons_ant: Langton's ant, which wanders chaotically for about 10,000 steps before it builds a periodic highway.

ant colony optimization

• Ant colony optimization on a road network ant_colony_optimization: Ants that find the shortest path between two points of a Delaunay road network, round after round.

anti-angiogenic therapy

• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.

antiangiogenic therapy

• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.

antibody

• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.

anticipation

• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.

antigenicity

• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.

ants

• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.

aquaculture

• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.

arbitrage

• Random-walk traders with arbitrage arbitrage_traders: Schwichtenberg's diffusion of a good by random-walk traders, minimally coupled to exchange rates.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

area preserving

• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
• Fermi-Ulam map fermi_ulam_map: The Fermi-Ulam map of a ball bouncing between a fixed wall and a wall that oscillates in time.

arms race

• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.

arnold tongues

• Circle circle_map: The sine circle map θ' = θ + Ω - K / (2π) sin 2πθ (mod 1), which locks into Arnold tongues.

artificial life

• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.

assembly theory

• Assembly growth along the assembly index assembly_growth: The numbers of objects at successive assembly indices, each discovered from the level below.
• Undirected and directed polymer assembly pool polymer_assembly_pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.

asset

• Correlated geometric Brownian motions correlated_geometric_brownian_motion: Two assets whose prices follow geometric Brownian motions driven by correlated Wiener processes.
• Geometric Brownian motion geometric_brownian_motion: An asset whose price follows geometric Brownian motion.

asset pricing

• Brock-Hommes heuristic switching brock_hommes: Brock and Hommes' asset market with traders switching between fundamentalists and trend chasers.

associative memory

• Asymmetric diluted Hopfield network diluted_hopfield: The asymmetric diluted Hopfield network, which recalls a stored pattern below its memory capacity.
• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.
• Hopfield network hopfield: A Hopfield network that stores patterns with Hebb's rule and recalls the first from a corrupted copy.

assortativity

• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.

asynchronous

• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.

atmosphere

• Lorenz-84 lorenz_84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.
• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.

attack

• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.

attractor

• Chua double and multiple scrolls chua: Chua's circuit, whose double-scroll or multiscroll attractor is chaotic.
• Hopfield network hopfield: A Hopfield network that stores patterns with Hebb's rule and recalls the first from a corrupted copy.

attractors

• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Three-gene Boolean network three_gene_boolean_network: The three-gene Boolean network of Raman, updated synchronously.

autocatalysis

• Brusselator ODE brusselator_ode: The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
• Cubic autocatalysis cubic_autocatalysis: Cubic autocatalysis A + 2B -> 3B with the decay B -> C in a continuously fed reactor.
• Hypercycles hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.

autocatalytic set

• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.

autocatalytic sets

• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.

autoregressive

• Random walk with autoregressive velocity random_walk_with_memory: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.

avalanches

• Bak-Tang-Wiesenfeld sandpile sandpile: The Bak-Tang-Wiesenfeld sandpile on a lattice with open edges, started from random stable heights.
• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• Manna sandpile manna_sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

backpropagation through time

• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.

bacteria

• Run-and-tumble run_and_tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.

balancing feedback

• Cooling coffee cooling_coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).

bang-bang

• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.
• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.

banks

• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.
• Fractional-reserve banking fractional_reserve_banking: Customers and banks whose lending of excess reserves multiplies the base money by 1 / ratio.

barter

• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.
• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.
• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.

basins

• Newton-Raphson map newton_raphson_map: Newton's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.

batch culture

• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.

batteries

• Faraday-disk charger faraday_disk: A battery charger driven by a Faraday disk generator whose own current strengthens its field.

bernoulli map

• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.

bernoulli shift

• Bernoulli doubling map doubling_map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.

betweenness

• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.

bid rent

• Von Thünen land use von_thunen: A square plain around one market at its centre, whose every cell grows the crop of highest bid rent.

bifurcation

• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.
• Hopf hopf_form: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
• Saddle-node saddle_node_form: The saddle-node normal form x' = μ - x², whose two equilibria ±√μ meet and vanish at μ = 0.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Takens-Bogdanov takens_bogdanov_form: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.

bifurcation diagram

• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.

bijective

• Power-series automaton power_series_automaton: The automaton τ(x)(n) = x(n) - t x(n + 1) over the ring K[[t]] of formal power series on a ring.

billiard

• Circular billiard circular_billiard: A particle moving freely inside a circle and reflecting specularly off its wall.
• Dynamical billiards of Sinai, mushroom, and open three-disk shapes billiards: A particle moving freely on a billiard table and reflecting specularly off its walls.
• Fermi-Ulam map fermi_ulam_map: The Fermi-Ulam map of a ball bouncing between a fixed wall and a wall that oscillates in time.

binomial

• Two-phase zone exchange two_phase_zone_exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.

biochemical systems theory

• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.

biodiversity

• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.

bioenergetics

• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.

biomodels

• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.

bipartite

• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.

bipartite network

• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.

birth-death

• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.

birth-death process

• Birth-death process and M/M/1 queue birth_death_queue: Independent M/M/1 queues, birth-death processes with constant arrival and service rates.
• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.

bistability

• Bistability on a catalyst pellet catalyst_pellet: The surface concentration of a catalyst pellet, bistable through Langmuir-Hinshelwood kinetics.
• Chemical Langevin equation of the Schlögl reaction chemical_langevin_equation: The chemical Langevin equation of Schlögl's bistable reaction, whose noise switches between two states.
• Cubic autocatalysis cubic_autocatalysis: Cubic autocatalysis A + 2B -> 3B with the decay B -> C in a continuously fed reactor.
• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Genetic toggle switch toggle_switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• SIS on hypergraphs with heterogeneous exposure and a nonlinear infection kernel hypergraph_exposure_sis: The SIS contagion of St-Onge et al. in the environments of a hypergraph, with a nonlinear kernel.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.
• Stommel thermohaline boxes stommel: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Zero-dimensional energy balance with ice albedo energy_balance: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.

black holes

• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.

blinking traps

• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.

blocked states

• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.

boltzmann equation

• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.

bone marrow

• Bone-marrow and blood cells with blasts and cancer stem cells bone_marrow_blood_leukaemia: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.

boolean network

• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Kauffman random Boolean networks random_boolean_network: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Probabilistic Boolean network probabilistic_boolean_network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.
• Three-gene Boolean network three_gene_boolean_network: The three-gene Boolean network of Raman, updated synchronously.
• Yeast cell-cycle network yeast_cell_cycle: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.

boom and bust

• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.

boundary

• Ideal gas ideal_gas: An ideal gas of non-interacting molecules in a reflecting box.

bounded confidence

• Deffuant bounded confidence deffuant: The Deffuant bounded confidence system, whose opinions settle into about 1 / (2 * confidence) clusters.
• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.

bounded rationality

• Brock-Hommes heuristic switching brock_hommes: Brock and Hommes' asset market with traders switching between fundamentalists and trend chasers.
• El Farol el_farol: Arthur's El Farol bar problem, whose attendance self-organizes around the capacity.
• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.

braess paradox

• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.

branching process

• Galton-Watson branching galton_watson: A Galton-Watson branching process with Poisson offspring, one generation per step.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.

breaking

• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

breast cancer

• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.

bristles

• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.

brownian motion

• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Brownian particle near a wall brownian_near_wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.
• Inertial and overdamped Brownian motion brownian_motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.
• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Optical trap including oscillating and blinking traps optical_trap: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.
• Overdamped particle in a tilted periodic potential tilted_periodic_potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.
• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.

bullwhip effect

• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.

bump

• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.

bursting

• Izhikevich izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.

bursts

• Barabási priority queues priority_queues: The Barabási priority queue, whose waiting times follow a power law as the choice becomes greedy.
• Negative autoregulation and bursty gene expression negative_autoregulation: A gene that represses its own transcription and makes its protein in bursts.

business cycle

• Multiplier-accelerator multiplier_accelerator: Samuelson's multiplier-accelerator, Y_{t+1} = c (1 + v) Y_t - c v Y_{t-1} + G.
• Toy firm investment toy_firm_investment: Heterogeneous firms that invest out of past profits and sell at random prices, with entry and exit.

busy beaver

• Turing machine (three-state busy beaver) turing_machine: A Turing machine, a head with an internal state that reads, writes, and moves on a tape of symbols.

canards

• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.

cancer stem cells

• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.

capital

• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.

car following

• Car following with delay delayed_car_following: A chain of cars that react to the speed difference to the car ahead after a reaction time.
• Intelligent driver system intelligent_driver_system: The intelligent driver system of cars on a ring road.
• Langevin car-following toward a density-dependent speed langevin_traffic: Cars on a ring road whose speeds relax with noise towards a desired speed set by the local density.
• Optimal velocity optimal_velocity: The optimal velocity system of cars on a ring road, where uniform flow breaks into stop-and-go waves.

carbon cycle

• Rothman excitable carbon cycle carbon_cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.

carbon-13 flux analysis

• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.

cardiac

• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.

cardiovascular

• Lumped cardiovascular system cardiovascular: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.

carrying capacity

• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.

cascade

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.
• Cascade classifier cascade_classifier: A cascade classifier, a column of stages that separates a narrow size class of particles by an air flow.
• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
• Cascading sample-space-reducing process cascading_sample_space_reducing: Cascading sample-space-reducing processes, whose visits follow the power law i^(-mu) for any mu.
• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.

cascades

• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Watts threshold cascades watts_cascades: The Watts threshold system of global cascades started by a few random seeds.

catalysis

• Bistability on a catalyst pellet catalyst_pellet: The surface concentration of a catalyst pellet, bistable through Langmuir-Hinshelwood kinetics.

celestial mechanics

• Satellite transfer satellite_transfer: A satellite moved between Kepler orbits of equal angular momentum by radial thrust.

cell communication

• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.

cell complex

• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.
• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

cell cycle

• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Cell-cycle compartments with killing, blocking, and recruiting agents cell_cycle_chemotherapy: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Yeast cell-cycle network yeast_cell_cycle: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.

cell fate

• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.

cell quota

• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.

cell sorting

• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.

cellular automaton

• Automata on the Grigorchuk group grigorchuk_automaton: The majority automaton on a ball of the Grigorchuk group, a group of intermediate growth.
• Automata on the free group free_group_automaton: The majority automaton on a ball of the free group, a tree in which every cell follows its neighbours.
• Baumslag-Solitar groups baumslag_solitar_automaton: The majority automaton on a ball of the Baumslag-Solitar group BS(m, n), of exponential growth.
• Cyclic cellular automaton cyclic_cellular_automaton: Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
• Heisenberg heisenberg_automaton: The majority automaton on a ball of the discrete Heisenberg group, of polynomial growth of degree 4.
• Involutive automaton involutive_automaton: A binary automaton on a ring that is its own inverse, with the minimal memory set {-1, 0, ..., n}.
• Lamplighter lamplighter_automaton: The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.
• Linear automata over rings linear_ring_automaton: The discrete Laplacian as a linear automaton on a ring, over the field of modulus elements.
• Muller's pre-injective non-surjective automaton on F2 muller_automaton: Muller's automaton on the free group F2, pre-injective but not surjective.
• Nilpotent automaton of class n nilpotent_automaton: A binary automaton on a ring that is nilpotent of class n: n steps send every configuration to 0.
• Q2R reversible Ising automaton q2r: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.

cellular potts

• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.

census

• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.

centrality

• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.
• PageRank iterations on networks pagerank: The PageRank iteration r_i' = (1 - d) / n + d Σ_j A_ij r_j / k_j on an undirected network.

chaos

• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Baker baker_map: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Bernoulli doubling map doubling_map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.
• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
• Cat map cat_map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.
• Chua double and multiple scrolls chua: Chua's circuit, whose double-scroll or multiscroll attractor is chaotic.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Coupled Colpitts oscillators coupled_colpitts: Two detuned chaotic Colpitts oscillators coupled through the difference of their second variables.
• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.
• Coupled map lattices on networks coupled_map_lattices_on_networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Coupled soil and deep temperature maps soil_temperature: The coupled maps of the soil surface temperature and the deeper soil temperature.
• Cubic chaotic map and its chaotic walk cubic_chaotic_map: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Duffing duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• Dynamical billiards of Sinai, mushroom, and open three-disk shapes billiards: A particle moving freely on a billiard table and reflecting specularly off its walls.
• Elementary automata elementary_automaton: An elementary cellular automaton by its Wolfram number, started from a single live cell by default.
• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Fractional Lorenz fractional_lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
• Generalized tent map skew_tent_map: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.
• Gissinger geodynamo geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.
• Hyperchaotic four-dimensional Rössler flow hyperchaotic_rossler: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
• Hénon map henon_map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
• Hénon-Heiles henon_heiles: The Hénon-Heiles system of a star in a galactic potential, whose motion is chaotic at higher energies.
• Intermittency maps intermittency_map: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Langford attractor langford: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
• Logistic map logistic_map: The logistic map x_{t+1} = r x_t (1 - x_t), which reaches chaos through period doubling as r grows.
• Lorenz lorenz: The Lorenz-63 system of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
• Lorenz-84 lorenz_84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.
• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.
• Lozi map lozi_map: The Lozi map x' = 1 + y - a |x|, y' = b x, the piecewise linear Hénon map with a strange attractor.
• Mackey-Glass mackey_glass: The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.
• Memristor oscillator memristor_oscillator: The simplest memristor oscillator: a capacitor, an inductor, and a memristor in series.
• N-body and solar system n_body: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.
• Nuclear quadrupole vibrations nuclear_quadrupole: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Ricker map of overcompensating generations ricker: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.
• Rössler rossler: The Rössler system, a chaotic flow with one folded band for its standard parameters.
• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
• Rössler-driven Lorenz and generalized synchronization rossler_driven_lorenz: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.
• Sprott jerk sprott_jerk: Sprott's jerk equation x''' = -A x'' - x' + |x| - 1, chaotic near A = 0.6.
• Standard standard_map: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Tent tent_map: The tent map x' = μ min(x, 1 - x), chaotic with Lyapunov exponent ln μ for μ above 1.
• Thomas cyclically symmetric attractor thomas: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.
• Towel maps towel_map: Rössler's towel map, a three-dimensional map with two positive Lyapunov exponents.

chaos game

• Iterated function systems by the chaos game chaos_game: Points that draw the attractor of an iterated function system by the chaos game.

chartists

• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.

checkpoint inhibitor

• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.

chemical Langevin equation

• Chemical Langevin equation of the Schlögl reaction chemical_langevin_equation: The chemical Langevin equation of Schlögl's bistable reaction, whose noise switches between two states.
• Chemical Langevin equation of the enzyme network chemical_langevin_network: The chemical Langevin equation of the enzyme reaction E + S <-> C -> E + P.

chemotaxis

• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.

chemotherapy

• Acquired and induced resistance with phenotype switching drug_resistance_switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.
• Cell-cycle compartments with killing, blocking, and recruiting agents cell_cycle_chemotherapy: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Normal, tumour, and immune cells under chemotherapy de_pillis_radunskaya: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Optimal chemotherapy dosing optimal_chemotherapy: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.
• Primary and secondary sites with a migration delay under chemotherapy delayed_metastasis: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.
• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.
• Resistance subpopulations under three drugs multidrug_resistance: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.
• Resting and hunting immune cells with a conversion delay under chemotherapy delayed_resting_hunting_chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.
• Skipper-Schabel-Wilcox log-kill skipper_log_kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.
• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.

chimera

• Kuramoto-Sakaguchi chimera kuramoto_sakaguchi_chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.

chiral phase

• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.

chirality

• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Chiral active particles chiral_active_particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.

circadian rhythm

• Drosophila circadian clock circadian_clock: Goldbeter's system of the Drosophila circadian clock, PER that represses its own transcription.

circuit

• Chua double and multiple scrolls chua: Chua's circuit, whose double-scroll or multiscroll attractor is chaotic.
• Circuits under periodic impulses impulsive_circuits: An RLC circuit driven by a periodic train of current impulses, L C x'' + R C x' + x = A Σ δ(t - k T).
• Coupled Colpitts oscillators coupled_colpitts: Two detuned chaotic Colpitts oscillators coupled through the difference of their second variables.
• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.
• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Memristor oscillator memristor_oscillator: The simplest memristor oscillator: a capacitor, an inductor, and a memristor in series.
• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.
• RLC circuit rlc_circuit: A series RLC circuit written as Kirchhoff's voltage law, L q'' + R q' + q / C = V cos(ω t).
• Tunnel-diode oscillator tunnel_diode_oscillator: A tunnel-diode oscillator: an inductor and a capacitor in parallel with a negative-resistance diode.

circular motion

• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.

circulation

• Lumped cardiovascular system cardiovascular: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.

clearing

• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.

clubs

• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.

clustering

• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.
• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.

clusters

• Continuum percolation continuum_percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Deffuant bounded confidence deffuant: The Deffuant bounded confidence system, whose opinions settle into about 1 / (2 * confidence) clusters.
• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.
• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Social impact theory social_impact: The social impact system on a full grid, with random persuasiveness and supportiveness.

co-simulation

• Active walkers and trail formation active_walkers: Active walkers who walk between the corners of a green and wear a system of trails into it.
• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Harvesting farmers harvesting: A socio-ecological system of farmer agents harvesting a logistically regrowing resource.
• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.
• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• Pheromone foraging pheromone_foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.
• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.

coagulation

• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.

coalitions

• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.

coarsening

• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.

codimension two

• Takens-Bogdanov takens_bogdanov_form: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.

coevolution

• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Tangled Nature tangled_nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.

cognition

• Cognitive consistency agents cognitive_consistency: Agents whose attitudes spread by conversation and settle in a belief network that seeks consistency.

cognitive systems

• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.

cohesion

• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.

collapse

• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.

collective decision

• Binary ant decision binary_ant_decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.

collisions

• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.

collocation

• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.

colloidal molecule

• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.

colored noise

• Active Ornstein-Uhlenbeck particles active_ornstein_uhlenbeck_particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Langevin equation with colored noise colored_noise_langevin: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.

combination therapy

• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.
• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.

combinatorial evolution

• Arthur-Polak technology evolution arthur_polak: Arthur and Polak's evolution of technology, which builds logic functions from NAND gates.
• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.
• Combinatorial creative phase transition creative_phase_transition: The diversity of a combinatorial system, which jumps from sparse to full above a critical rule density.

commodity cycle

• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.

commons

• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.
• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.

communities

• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.

community flux balance

• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.

commuting

• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.

compartments

• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Cell-cycle compartments with killing, blocking, and recruiting agents cell_cycle_chemotherapy: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• SIR sir: An epidemic that moves individuals from susceptible to infected to recovered.
• SIR on a network as an agent system network_sir: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.
• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS sis: An infection without immunity, where the recovered become susceptible again and the disease is endemic.
• Two-compartment pharmacokinetics two_compartment_pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.
• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

competition

• Competition competition: Two species that compete for the same resources, which coexist or exclude each other.
• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Fukai-Tanaka winner-take-all network winners_take_all: A winner-take-all network of units with self-limited growth and mutual inhibition.
• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and tumour competition under radiation radiotherapy_competition: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.
• Normal, tumour, and immune cells under chemotherapy de_pillis_radunskaya: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Uninfected and infected tumour cells under an oncolytic virus oncolytic_virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.

competitive exclusion

• Consumer-resource models of substitutable and essential resources consumer_resource: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.

complex

• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).
• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.

computation

• Turing machine (three-state busy beaver) turing_machine: A Turing machine, a head with an internal state that reads, writes, and moves on a tape of symbols.

concurrency

• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.
• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.

condensation

• Zero-range process zero_range_process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

conductance

• Hodgkin-Huxley hodgkin_huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.

configuration model

• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.

connectance

• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.

consensus

• Galam majority galam_majority: The Galam majority system, where random groups adopt their majority until one opinion wins.
• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.
• Naming game naming_game: The minimal naming game, where agents with empty inventories agree on one word.
• Nonlinear three-body consensus on temporal hypergraphs hypergraph_consensus: The nonlinear three-body consensus system on a random hypergraph of triangles that switch on and off.
• Sznajd sznajd: The Sznajd system, where agreeing pairs of agents convince their neighbours.

conservation

• Diffusion automaton diffusion_automaton: A cellular automaton of diffusion, c' = c + r Σ (c_neighbour - c), started from a square of ones.
• FHP lattice gas fhp_lattice_gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.

conservation law

• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

conservative

• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.

conserved

• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.

conserved noise

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.

consumer choice

• Fair-trade food choice with peer influence fair_food: The choice between fair-trade and regular food of consumers who influence their neighbours.

consumer-resource

• Consumer-resource models of substitutable and essential resources consumer_resource: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.

contact network

• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.

contagion

• Calm and nervous rats rats: A colony of calm and nervous rats, in which nervousness spreads by contact and fades on its own.
• DebtRank cascades debtrank: The DebtRank cascade of distress from a set of banks through their interbank loans.
• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.
• SIS on hypergraphs with heterogeneous exposure and a nonlinear infection kernel hypergraph_exposure_sis: The SIS contagion of St-Onge et al. in the environments of a hypergraph, with a nonlinear kernel.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.

containers

• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.

context-free

• Context-free subshift context_free_subshift: A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.

continuous automaton

• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.

continuous energy

• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

continuous space

• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions boids: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.
• Obesity system obesity: The obesity system of adults who eat near home and work, one step per day.
• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.

contract curve

• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.

contractile

• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.

contrarians

• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.

control

• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.
• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.
• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.
• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.
• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.
• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.
• Satellite transfer satellite_transfer: A satellite moved between Kepler orbits of equal angular momentum by radial thrust.
• Thermostat thermostat: A room heated by a furnace under a thermostat while heat leaks to the outside.

convection

• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.

convolution

• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.
• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.

cooperation

• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.
• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.
• Iterated prisoner's dilemma tournament prisoners_dilemma_tournament: The evolution of strategies of the iterated prisoner's dilemma from random genomes.
• Public goods game public_goods_game: The spatial public goods game in groups of five neighbours, with Fermi imitation.
• Public goods game on hypergraphs hypergraph_public_goods: The public goods game on a hypergraph, played in every hyperedge, with Fermi imitation.
• Spatial prisoner's dilemma with best-neighbour and Fermi imitation spatial_prisoners_dilemma: The spatial prisoner's dilemma of Nowak and May, with best-neighbour or Fermi imitation.
• Two-player games on links and three-body games on 2-simplices simplicial_social_dilemmas: The social dilemmas of Guo et al. on the links and the 2-simplices of a simplicial complex.

cooperative game theory

• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.

copying

• Vertex-copying network growth vertex_copying: Vertex-copying growth, whose new nodes copy the edges of existing nodes and make a heavy-tailed degree.

correlated noise

• Correlated geometric Brownian motions correlated_geometric_brownian_motion: Two assets whose prices follow geometric Brownian motions driven by correlated Wiener processes.
• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.

cosmology

• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.

costs

• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.

counting process

• Bernoulli process bernoulli_process: Independent Bernoulli processes, one success or failure per step, whose success counts are binomial.
• Poisson process poisson_process: Independent Poisson processes, counters that grow by one at exponentially distributed waiting times.

coupled maps

• Coupled soil and deep temperature maps soil_temperature: The coupled maps of the soil surface temperature and the deeper soil temperature.
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.

coupling tensor

• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.

covid

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.

credit

• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.

crime

• Brothel movement system trafficking: The brothel movement system, where victims moved between brothels leave traces in bills and train trips.

criminality

• Criminality and security criminality: The kinetic system of criminality in a society and the security forces that act against it.

crises

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.

critical load

• Binary perceptron and random K-SAT perceptron_satisfiability: A binary perceptron that must store random patterns, revealed one per step, as cardinality constraints.

critical slowing down

• Landau mean-field relaxation landau_relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.

criticality

• Galton-Watson branching galton_watson: A Galton-Watson branching process with Poisson offspring, one generation per step.
• Kauffman random Boolean networks random_boolean_network: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.
• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.

cross-feeding

• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.

crowds

• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.

culture

• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.

current

• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Overdamped particle in a tilted periodic potential tilted_periodic_potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.

cyclic dominance

• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.

cyclic symmetry

• Thomas cyclically symmetric attractor thomas: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.

cytokines

• Kirschner-Panetta system with TGF-beta and siRNA sirna_tgf_beta: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.

dark energy

• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.

data assimilation

• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.

dde

• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Ikeda delay equation ikeda: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.
• Primary and secondary sites with a migration delay under chemotherapy delayed_metastasis: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.
• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.
• Resting and hunting immune cells with a conversion delay under chemotherapy delayed_resting_hunting_chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.

decoupling

• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

default cascade

• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.

deformable particles

• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.

degree distribution

• Kronecker graphs kronecker_graph: The Kronecker graph of two graphs, whose adjacency matrix is the Kronecker product of theirs.
• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.

degree sequence

• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.

delay

• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Bone-marrow and blood cells with blasts and cancer stem cells bone_marrow_blood_leukaemia: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.
• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Car following with delay delayed_car_following: A chain of cars that react to the speed difference to the car ahead after a reaction time.
• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Delayed Kuramoto delayed_kuramoto: Kuramoto phase oscillators coupled all to all through signals that arrive after a delay.
• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• ENSO delay oscillator enso: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).
• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Hutchinson hutchinson: Hutchinson's delayed logistic equation, which oscillates around K when r * tau exceeds π/2.
• Ikeda delay equation ikeda: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Light-sensitive robots with sensory delay light_seeking_robots: Light-sensitive robots in a circular well, whose sensory delay decides whether they gather in the light.
• Mackey-Glass mackey_glass: The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Primary and secondary sites with a migration delay under chemotherapy delayed_metastasis: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.
• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.
• Resting and hunting immune cells with a conversion delay under chemotherapy delayed_resting_hunting_chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.
• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.
• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.
• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.

deletion

• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.

demand

• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.

dendritic cells

• Oncolytic adenovirus with T cells and dendritic-cell injection oncolytic_virus_dendritic: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.

detailed balance

• Discrete master equation discrete_master_equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.

deterministic

• Employment takeover employment_takeover: The employment takeover system on a periodic checkerboard, with half of each kind coloured at random.
• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.

diauxie

• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.

differential adhesion

• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.

diffusion

• Active Brownian particles active_brownian_particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.
• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Cubic chaotic map and its chaotic walk cubic_chaotic_map: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
• Diffusion network_diffusion: Diffusion on a network, du_i/dt = D Σ_j A_ij (u_j - u_i), which spreads a unit mass to every node.
• Diffusion automaton diffusion_automaton: A cellular automaton of diffusion, c' = c + r Σ (c_neighbour - c), started from a square of ones.
• Fixed-line phone adoption phone_adoption: Hamill and Gilbert's adoption of fixed-line phones by British households from 1951.
• Heat diffusion heat_diffusion: Heat diffusing on a plate from a hot square in its middle, du/dt = D laplacian(u).
• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.
• Inertial and overdamped Brownian motion brownian_motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.
• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Persistent random walk persistent_random_walk: Walkers on the integers that keep the direction of their last step with a probability.
• Random walks random_walk: Independent walkers on the integers that step at random times, a continuous-time random walk.
• Random-walk traders with arbitrage arbitrage_traders: Schwichtenberg's diffusion of a good by random-walk traders, minimally coupled to exchange rates.

digital market

• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.

dimension reduction

• Ott-Antonsen reduction ott_antonsen: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.

directed percolation

• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.

discrete choice

• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.

discrete map

• Adaptrode synapse adaptrode: An Adaptrode synapse driven by a train of input pulses that stops after a while.
• Asymmetric diluted Hopfield network diluted_hopfield: The asymmetric diluted Hopfield network, which recalls a stored pattern below its memory capacity.
• Baker baker_map: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Bernoulli doubling map doubling_map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.
• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.
• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
• Cat map cat_map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.
• Circle circle_map: The sine circle map θ' = θ + Ω - K / (2π) sin 2πθ (mod 1), which locks into Arnold tongues.
• Cobweb cobweb: A market whose price converges to or oscillates around equilibrium as supply lags the price.
• Combinatorial creative phase transition creative_phase_transition: The diversity of a combinatorial system, which jumps from sparse to full above a critical rule density.
• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
• Coupled soil and deep temperature maps soil_temperature: The coupled maps of the soil surface temperature and the deeper soil temperature.
• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.
• Cubic chaotic map and its chaotic walk cubic_chaotic_map: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• Echo state network echo_state_network: An echo state network, a reservoir of leaky tanh units with random recurrent weights driven by a sine.
• Ehrenfest urn system of two boxes ehrenfest_urn: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.
• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.
• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.
• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• Exchange rates in international trade exchange_rates: Hamill and Gilbert's staged adjustment of trade and the exchange rate to a rise in home prices.
• Fermi-Ulam map fermi_ulam_map: The Fermi-Ulam map of a ball bouncing between a fixed wall and a wall that oscillates in time.
• Fibonacci rabbits fibonacci_rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.
• Generalized tent map skew_tent_map: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.
• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.
• Hidden Markov source hidden_markov_source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.
• Hénon map henon_map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
• Intermittency maps intermittency_map: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
• Intrinsic-instability populations intrinsic_instability: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
• Iterated function systems by the chaos game chaos_game: Points that draw the attractor of an iterated function system by the chaos game.
• Kauffman random Boolean networks random_boolean_network: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.
• Logistic map logistic_map: The logistic map x_{t+1} = r x_t (1 - x_t), which reaches chaos through period doubling as r grows.
• Lozi map lozi_map: The Lozi map x' = 1 + y - a |x|, y' = b x, the piecewise linear Hénon map with a strange attractor.
• Markov source markov_source: A Markov source, a sequence of symbols in which each symbol depends on the one before it.
• Multiplier-accelerator multiplier_accelerator: Samuelson's multiplier-accelerator, Y_{t+1} = c (1 + v) Y_t - c v Y_{t-1} + G.
• Newton-Raphson map newton_raphson_map: Newton's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.
• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.
• Nicholson-Bailey nicholson_bailey: Hosts and parasitoids in discrete generations, whose equilibrium is unstable to growing oscillations.
• PageRank iterations on networks pagerank: The PageRank iteration r_i' = (1 - d) / n + d Σ_j A_ij r_j / k_j on an undirected network.
• Persistent random walk persistent_random_walk: Walkers on the integers that keep the direction of their last step with a probability.
• Probabilistic Boolean network probabilistic_boolean_network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Random walk with autoregressive velocity random_walk_with_memory: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.
• Ricker map of overcompensating generations ricker: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.
• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.
• Standard standard_map: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.
• Stock-flow consistent monetary economy stock_flow_consistent_economy: A stock-flow consistent economy, a closed circuit of money between households, firms, and a bank.
• Tent tent_map: The tent map x' = μ min(x, 1 - x), chaotic with Lyapunov exponent ln μ for μ above 1.
• Three-gene Boolean network three_gene_boolean_network: The three-gene Boolean network of Raman, updated synchronously.
• Towel maps towel_map: Rössler's towel map, a three-dimensional map with two positive Lyapunov exponents.
• Yeast cell-cycle network yeast_cell_cycle: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.

discrete time

• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

discrete velocities

• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.

disorder

• Binary perceptron and random K-SAT perceptron_satisfiability: A binary perceptron that must store random patterns, revealed one per step, as cardinality constraints.
• Continuum percolation continuum_percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Diffusion in a speckle field diffusion_in_a_speckle_field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.
• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Random energy system random_energy_system: The random energy system, 2^N quenched Gaussian energies of variance N / 2 at a temperature.
• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.

dispersal

• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.

distribution

• Criminality and security criminality: The kinetic system of criminality in a society and the security forces that act against it.
• Immune competition immune_competition: The kinetic system of the competition between tumour cells and immune cells.
• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.
• Kinetic opinion formation kinetic_opinion: The kinetic system of opinion formation over a discrete grid of opinions.
• Kinetic wealth and welfare dynamics kinetic_wealth_welfare: The kinetic theory of wealth classes, with a welfare policy when the tax is positive.

divergence entropy

• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.

diversity

• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.

dna computing

• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.

dose response

• Linear-quadratic cell survival and biologically effective dose lq_cell_survival: The linear-quadratic survival of cells after a total dose, with its biologically effective dose.

dosing

• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Cumulative drug toxicity drug_toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Normal and tumour competition under radiation radiotherapy_competition: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.
• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Skipper-Schabel-Wilcox log-kill skipper_log_kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.
• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.

double auction

• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.
• Zero-intelligence double auction double_auction: Gode and Sunder's double auction of zero-intelligence traders, a step per trading period.

double well

• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.

drive-response

• Rössler-driven Lorenz and generalized synchronization rossler_driven_lorenz: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.

driven

• Pulsed electrochemical reactor pulsed_electrochemical_reactor: A stirred electrochemical reactor whose electrode is driven by a train of current pulses.

drug resistance

• Acquired and induced resistance with phenotype switching drug_resistance_switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.
• Resistance subpopulations under three drugs multidrug_resistance: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.

duplication

• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.

dynamic aperture

• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.

dynamic flux balance

• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.

dynamic range

• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.

early stage

• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.

early warning

• Intrinsic-instability populations intrinsic_instability: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.

early warning signals

• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.

earthquakes

• Burridge-Knopoff spring-block fault burridge_knopoff: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Olami-Feder-Christensen earthquakes: The Olami-Feder-Christensen earthquake system with open edges, started from random forces.

echo chambers

• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.

echo state

• Echo state network echo_state_network: An echo state network, a reservoir of leaky tanh units with random recurrent weights driven by a sine.

ecosystems

• Tangled Nature tangled_nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.

edge contraction

• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.

eigenfunctions

• Quadratic system with a slow manifold and a finite Koopman embedding koopman_slow_manifold: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.

el farol

• El Farol el_farol: Arthur's El Farol bar problem, whose attendance self-organizes around the capacity.
• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.

electrochemistry

• Pulsed electrochemical reactor pulsed_electrochemical_reactor: A stirred electrochemical reactor whose electrode is driven by a train of current pulses.

electromagnet

• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.

electromagnetism

• RLC circuit rlc_circuit: A series RLC circuit written as Kirchhoff's voltage law, L q'' + R q' + q / C = V cos(ω t).
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.

electromechanics

• Faraday-disk charger faraday_disk: A battery charger driven by a Faraday disk generator whose own current strengthens its field.
• Regenerative braking regenerative_braking: A car braked by a Faraday disk generator on its wheel, which charges a supercapacitor.

elementary

• Elementary automata elementary_automaton: An elementary cellular automaton by its Wolfram number, started from a single live cell by default.
• Noisy automata noisy_automata: An elementary cellular automaton whose outputs are flipped at random with a small probability.

elementary flux modes

• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.

elementary metabolite units

• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.

elliptic umbilic

• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.

emergence

• Ajani's Choice ajanis_choice: Chesney's Ajani's Choice, migrant farm workers and employers among whom underpayment emerges.
• Game of Life game_of_life: Conway's Game of Life, or another life-like rule, on a torus.
• Game of Life on a rewired small-world network small_world_life: Conway's Game of Life on a small-world network rewired from the Moore lattice of a torus.
• Langton's ant langtons_ant: Langton's ant, which wanders chaotically for about 10,000 steps before it builds a periodic highway.

emotions

• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.

employment

• Employment takeover employment_takeover: The employment takeover system on a periodic checkerboard, with half of each kind coloured at random.

endemic

• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS sis: An infection without immunity, where the recovered become susceptible again and the disease is endemic.

energy

• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• Faraday-disk charger faraday_disk: A battery charger driven by a Faraday disk generator whose own current strengthens its field.
• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.
• Regenerative braking regenerative_braking: A car braked by a Faraday disk generator on its wheel, which charges a supercapacitor.
• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.

energy balance

• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.
• Zero-dimensional energy balance with ice albedo energy_balance: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.

energy cascade

• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.

energy depot

• Depot-pumped Brownian particles depot_brownian_particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).

energy harvesting

• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.

entropy

• Ehrenfest urn system of two boxes ehrenfest_urn: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.
• Mandelbrot least-effort word frequencies mandelbrot_least_effort: The word frequencies that minimise the mean cost of a language over its entropy, a power law of rank.

entropy rate

• Markov source markov_source: A Markov source, a sequence of symbols in which each symbol depends on the one before it.

enzyme kinetics

• Chemical Langevin equation of the enzyme network chemical_langevin_network: The chemical Langevin equation of the enzyme reaction E + S <-> C -> E + P.
• Michaelis-Menten as ODEs and as a jump process michaelis_menten: The enzyme reaction E + S <-> C -> E + P of Michaelis and Menten, as rate equations or as jumps.

epidemic threshold

• Activity-driven temporal networks activity_driven_epidemics: SIS spreading on an activity-driven temporal network, whose edges last one step.
• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.

epidemics

• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.

episims

• EpiSims city of people and locations episims_city: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.

equilibrium

• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.
• Static oligopoly equilibria oligopoly_equilibria: The equilibria of a Cournot duopoly and of a pricing game of firms with loyal customers and switchers.

equivariant

• Self-indexed shift over a group alphabet self_indexed_shift: The map τ(x)(g) = x(g · x(g)) over the group as alphabet, continuous and equivariant but no automaton.

eroding goals

• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.

error threshold

• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.

escape

• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.

escape time

• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).

euler-lagrange

• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.

evacuation

• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.

even shift

• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.

events

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Bouncing ball bouncing_ball: A ball that falls under gravity and bounces off the floor, losing speed by a coefficient of restitution.
• Circuits under periodic impulses impulsive_circuits: An RLC circuit driven by a periodic train of current impulses, L C x'' + R C x' + x = A Σ δ(t - k T).
• Circular billiard circular_billiard: A particle moving freely inside a circle and reflecting specularly off its wall.
• Dynamical billiards of Sinai, mushroom, and open three-disk shapes billiards: A particle moving freely on a billiard table and reflecting specularly off its walls.
• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.
• Integrate-and-fire integrate_and_fire: The leaky integrate-and-fire neuron, tau v' = -(v - rest) + R I, which spikes and resets at a threshold.
• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Izhikevich izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.
• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.
• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Mirollo-Strogatz pulse-coupled oscillators mirollo_strogatz: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Newton's cradle newtons_cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.

evolution

• Bit-string evolution on a fitness landscape bit_string_evolution: A population of bit strings that evolves on a random additive fitness landscape.
• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.
• Iterated prisoner's dilemma tournament prisoners_dilemma_tournament: The evolution of strategies of the iterated prisoner's dilemma from random genomes.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.
• Minority game and its evolutionary version minority_game: The minority game of an odd number of agents with two random strategies each.
• RobbyGA can collector and evolution by circumstance robby_ga: Mitchell's genetic algorithm that evolves strategies for Robby, a robot that collects cans in a room.
• Spatial prisoner's dilemma with best-neighbour and Fermi imitation spatial_prisoners_dilemma: The spatial prisoner's dilemma of Nowak and May, with best-neighbour or Fermi imitation.
• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.
• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

evolutionary algorithm

• Simulated annealing, genetic algorithms, and differential evolution evolutionary_optimisers: A population of candidates that searches a two-dimensional landscape, one generation per step.

evolutionary game

• Replicator dynamics and hawks and doves replicator_dynamics: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.
• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.

evolvability

• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.

exchange

• Boltzmann wealth exchange boltzmann_wealth: A Boltzmann wealth exchange system where every agent starts with one unit.
• Exchange with debt limit and with saving propensity kinetic_wealth_exchange: A kinetic wealth exchange system with a saving propensity or a debt limit.

exchange rate

• Exchange rates in international trade exchange_rates: Hamill and Gilbert's staged adjustment of trade and the exchange rate to a rise in home prices.
• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.

exchange rates

• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

excitability

• FitzHugh-Nagumo ODE fitzhugh_nagumo_ode: The FitzHugh-Nagumo neuron, the two-variable reduction of Hodgkin and Huxley's excitable membrane.
• Hodgkin-Huxley hodgkin_huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.
• Rothman excitable carbon cycle carbon_cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.

excitable

• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.
• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.

excitable media

• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Cyclic cellular automaton cyclic_cellular_automaton: Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
• FitzHugh-Nagumo spirals fitzhugh_nagumo: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• Hodgepodge machine hodgepodge: The hodgepodge machine, a system of a chemical reaction whose waves form spirals.

excitation-inhibition

• Wilson-Cowan wilson_cowan: The Wilson-Cowan system of interacting excitatory and inhibitory populations of neurons.

exclusion

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Competition competition: Two species that compete for the same resources, which coexist or exclude each other.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.

exploration

• Exploration and preferential return exploration_preferential_return: The exploration and preferential return model of human mobility of Song and coauthors.

explosive

• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.

explosive percolation

• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.

exponential growth

• Exponential growth observed at exponential random times exponential_growth_at_random_times: Exponential growth observed at exponentially distributed times, whose observed values follow a power law.

exposure

• SIS on hypergraphs with heterogeneous exposure and a nonlinear infection kernel hypergraph_exposure_sis: The SIS contagion of St-Onge et al. in the environments of a hypergraph, with a nonlinear kernel.

extinction

• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.
• Galton-Watson branching galton_watson: A Galton-Watson branching process with Poisson offspring, one generation per step.
• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.

extremal dynamics

• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.

extreme pathways

• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.

fair trade

• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.

feedback

• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.
• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.
• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.
• Thermostat thermostat: A room heated by a furnace under a thermostat while heat leaks to the outside.
• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

feedback control

• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.

fermi acceleration

• Fermi-Ulam map fermi_ulam_map: The Fermi-Ulam map of a ball bouncing between a fixed wall and a wall that oscillates in time.

fever

• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.

fibrinolysis

• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.

field

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Active walkers and trail formation active_walkers: Active walkers who walk between the corners of a green and wear a system of trails into it.
• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.
• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Burridge-Knopoff spring-block fault burridge_knopoff: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Decaying turbulence decaying_turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.
• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• Fisher-KPP fisher_kpp: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).
• FitzHugh-Nagumo spirals fitzhugh_nagumo: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Gray-Scott gray_scott: The Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.
• Heat diffusion heat_diffusion: Heat diffusing on a plate from a hot square in its middle, du/dt = D laplacian(u).
• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.
• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.
• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.
• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Pheromone foraging pheromone_foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Schnakenberg schnakenberg: The Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.
• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.
• Standing waves on a string standing_waves: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.
• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.
• Toner-Tu hydrodynamics toner_tu: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.
• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.
• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

final size

• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.

finite element

• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.

finite volume

• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

fire

• Drossel-Schwabl forest fire forest_fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Site percolation site_percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.

fire sales

• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.

firms

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.
• Toy firm investment toy_firm_investment: Heterogeneous firms that invest out of past profits and sell at random prices, with entry and exit.

fiscal policy

• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

fishery

• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.
• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.

fitness landscape

• Adaptive climbing in finite populations adaptive_climbing: A finite population that climbs a random fitness landscape by mutation and strong selection.
• Bit-string evolution on a fitness landscape bit_string_evolution: A population of bit strings that evolves on a random additive fitness landscape.
• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.
• NKCS coevolution on coupled NK landscapes nkcs: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Simulated annealing, genetic algorithms, and differential evolution evolutionary_optimisers: A population of candidates that searches a two-dimensional landscape, one generation per step.
• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.

fixation

• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.
• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.

fixed point

• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.

flocking

• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions boids: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Cucker-Smale flocking cucker_smale_flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Microscopic flocking with aligning binary collisions binary_collision_flocking: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.
• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.
• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
• Toner-Tu hydrodynamics toner_tu: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

fluctuations

• Two-phase zone exchange two_phase_zone_exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.

flux balance

• Flux balance, flux variability, and parsimonious FBA flux_balance: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.

fold

• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.

food security

• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.

food web

• Food webs food_webs: A food web where basal species grow logistically and consumers live on Lotka-Volterra predation.

forcing

• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Duffing duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.
• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Logistic growth with a varying capacity varying_capacity_logistic: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.
• Optical trap including oscillating and blinking traps optical_trap: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.
• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Stochastic resonance stochastic_resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.

forest

• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.

fractal

• Baker baker_map: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
• Cantor cantor_set: The Cantor set as the substitution of every interval by its two outer parts.
• Diffusion-limited aggregation diffusion_limited_aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Iterated function systems by the chaos game chaos_game: Points that draw the attractor of an iterated function system by the chaos game.
• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• Koch koch_curve: The Koch curve as the L-system F → F+F--F+F drawn by a turtle.
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).
• Newton-Raphson map newton_raphson_map: Newton's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.
• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.
• Sierpiński substitutions sierpinski_substitution: The Sierpiński triangle or carpet as the substitution of a polygon by scaled copies of itself.

fractal basins

• Magnetic pendulum magnetic_pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.

fractal network

• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

fractional

• Fractional Lorenz fractional_lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
• Fractional Van der Pol fractional_van_der_pol: The fractional Van der Pol oscillator D^alpha x = y, D^alpha y = mu (1 - x^2) y - x.
• Fractional damped oscillator fractional_oscillator: The fractional damped oscillator D^alpha x = v, D^alpha v = -omega^2 x - 2 gamma v.
• Fractional relaxation fractional_relaxation: Fractional relaxation D^alpha x = -lambda x, which decays as the Mittag-Leffler function E_alpha.

fractional noise

• Fractional Brownian motion fractional_brownian_motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.

fragmentation

• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.
• Fragmentation fragmentation: The random fragmentation of an object of integer size, whose fragments keep breaking into smaller ones.

frame dragging

• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.

free group

• Automata on the free group free_group_automaton: The majority automaton on a ball of the free group, a tree in which every cell follows its neighbours.
• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.
• Muller's pre-injective non-surjective automaton on F2 muller_automaton: Muller's automaton on the free group F2, pre-injective but not surjective.
• Self-indexed shift over a group alphabet self_indexed_shift: The map τ(x)(g) = x(g · x(g)) over the group as alphabet, continuous and equivariant but no automaton.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.

frequency clusters

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.

friction

• Burridge-Knopoff spring-block fault burridge_knopoff: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.

friendship paradox

• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.

fronts

• Fisher-KPP fisher_kpp: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).

frustration

• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.

fundamentalists

• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.

g protein

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.

game

• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.

game theory

• Replicator dynamics and hawks and doves replicator_dynamics: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.
• Static oligopoly equilibria oligopoly_equilibria: The equilibria of a Cournot duopoly and of a pricing game of firms with loyal customers and switchers.

game transitions

• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.

games

• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.
• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.
• Iterated prisoner's dilemma tournament prisoners_dilemma_tournament: The evolution of strategies of the iterated prisoner's dilemma from random genomes.
• Public goods game public_goods_game: The spatial public goods game in groups of five neighbours, with Fermi imitation.
• Public goods game on hypergraphs hypergraph_public_goods: The public goods game on a hypergraph, played in every hyperedge, with Fermi imitation.
• Spatial prisoner's dilemma with best-neighbour and Fermi imitation spatial_prisoners_dilemma: The spatial prisoner's dilemma of Nowak and May, with best-neighbour or Fermi imitation.
• Two-player games on links and three-body games on 2-simplices simplicial_social_dilemmas: The social dilemmas of Guo et al. on the links and the 2-simplices of a simplicial complex.

gamma oscillations

• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.

gap junctions

• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.

gas

• Ideal gas ideal_gas: An ideal gas of non-interacting molecules in a reflecting box.

gauge

• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.
• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

gauge field

• Random-walk traders with arbitrage arbitrage_traders: Schwichtenberg's diffusion of a good by random-walk traders, minimally coupled to exchange rates.
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.

gel

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.

gene expression

• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.

gene knock-out

• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.

gene regulation

• Drosophila circadian clock circadian_clock: Goldbeter's system of the Drosophila circadian clock, PER that represses its own transcription.
• Genetic toggle switch toggle_switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Kauffman random Boolean networks random_boolean_network: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Negative autoregulation and bursty gene expression negative_autoregulation: A gene that represses its own transcription and makes its protein in bursts.
• Probabilistic Boolean network probabilistic_boolean_network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Repressilator repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.
• Three-gene Boolean network three_gene_boolean_network: The three-gene Boolean network of Raman, updated synchronously.
• Yeast cell-cycle network yeast_cell_cycle: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.

gene therapy

• Kirschner-Panetta system with TGF-beta and siRNA sirna_tgf_beta: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Predator-prey tumour-immune system of TCR gene therapy tcr_gene_therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.

general circulation

• Lorenz-84 lorenz_84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.

general relativity

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.

generalized mass action

• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.

generalized synchronization

• Rössler-driven Lorenz and generalized synchronization rossler_driven_lorenz: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.

genetic algorithm

• Bit-string evolution on a fitness landscape bit_string_evolution: A population of bit strings that evolves on a random additive fitness landscape.
• RobbyGA can collector and evolution by circumstance robby_ga: Mitchell's genetic algorithm that evolves strategies for Robby, a robot that collects cans in a room.

genetic drift

• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.

genotype networks

• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.

geodesics

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.

geomagnetism

• Gissinger geodynamo geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.

giant component

• Erdos-Renyi and Watts-Strogatz graphs random_graph_ensembles: The construction of an Erdős-Rényi or a Watts-Strogatz random graph, one step of new edges at a time.
• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.

giant number fluctuations

• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.

gillespie

• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Generalized stochastic predator-prey stochastic_predator_prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.
• Michaelis-Menten as ODEs and as a jump process michaelis_menten: The enzyme reaction E + S <-> C -> E + P of Michaelis and Menten, as rate equations or as jumps.

glass

• Bouchaud trap system trap_system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.

glass transition

• Random energy system random_energy_system: The random energy system, 2^N quenched Gaussian energies of variance N / 2 at a temperature.

glioma

• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.

glucagon

• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.

glycolysis

• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.

golden mean

• Subshifts subshifts: A Markov chain on the golden mean shift, the configurations of a ring that never hold two adjacent ones.

governance

• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.

granular matter

• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.

graph laplacian

• Diffusion network_diffusion: Diffusion on a network, du_i/dt = D Σ_j A_ij (u_j - u_i), which spreads a unit mass to every node.

graphlet

• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.

gravity

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• N-body and solar system n_body: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Two-body orbit two_body_orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.

gravity system

• Gravity interaction system of urban scaling gravity_urban_scaling: The residents of a city at uniform positions in a disc, each interacting with the others as r^-gamma.

grazing

• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.

grid

• Agent-based Lotka-Volterra agent_based_lotka_volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.
• Belief diffusion with influentials influentials: The belief diffusion system with influentials who care passionately and weigh more with their neighbours.
• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.
• Bonabeau hierarchies bonabeau_hierarchies: The Bonabeau system, where a hierarchy of strengths emerges above a critical density of agents.
• Brothel movement system trafficking: The brothel movement system, where victims moved between brothels leave traces in bills and train trips.
• Carrying capacity of a meadow meadow_capacity: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.
• Fair-trade food choice with peer influence fair_food: The choice between fair-trade and regular food of consumers who influence their neighbours.
• Insider sabotage insider_sabotage: The insider sabotage system of employees whose risk of sabotage builds up over the days of a year.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.
• Schelling schelling: A Schelling segregation system on a grid with at most one resident per cell.
• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.
• Social impact theory social_impact: The social impact system on a full grid, with random persuasiveness and supportiveness.
• Stochastic Schelling stochastic_schelling: The stochastic Schelling system of agents moving between blocks, which segregates above beta = 2.
• Sugarscape sugarscape: The Sugarscape of Epstein and Axtell, foragers of random vision and metabolism on two sugar peaks.
• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.
• Wolf and sheep wolf_sheep: The wolf and sheep predation system with one grass patch per cell.

group

• Automata on the Grigorchuk group grigorchuk_automaton: The majority automaton on a ball of the Grigorchuk group, a group of intermediate growth.
• Automata on the free group free_group_automaton: The majority automaton on a ball of the free group, a tree in which every cell follows its neighbours.
• Baumslag-Solitar groups baumslag_solitar_automaton: The majority automaton on a ball of the Baumslag-Solitar group BS(m, n), of exponential growth.
• Heisenberg heisenberg_automaton: The majority automaton on a ball of the discrete Heisenberg group, of polynomial growth of degree 4.
• Lamplighter lamplighter_automaton: The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.
• Linear automata over rings linear_ring_automaton: The discrete Laplacian as a linear automaton on a ring, over the field of modulus elements.
• Muller's pre-injective non-surjective automaton on F2 muller_automaton: Muller's automaton on the free group F2, pre-injective but not surjective.
• Self-indexed shift over a group alphabet self_indexed_shift: The map τ(x)(g) = x(g · x(g)) over the group as alphabet, continuous and equivariant but no automaton.
• Subshifts subshifts: A Markov chain on the golden mean shift, the configurations of a ring that never hold two adjacent ones.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.

group alphabet

• Group-valued difference automaton group_difference_automaton: The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.

group formation

• El Farol with strategic group formation el_farol_coalitions: Collins' El Farol bar with strategic group formation, whose coalitions share their best strategy.

growth

• Barabási-Albert growth, with Bianconi-Barabási fitness barabasi_albert: Barabási-Albert growth, whose degrees follow a power law P(k) ~ k^-3 by preferential attachment.
• Capital stock capital_stock: Meadows' capital stock, grown by reinvested output and drained by depreciation.
• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.
• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.
• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.
• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.
• Simon-Yule growth simon_yule: The Simon system of a text that grows from one word, whose word counts follow a power law.
• Solow solow: An economy whose capital per worker approaches its steady state through saving and depreciation.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.
• Vertex-copying network growth vertex_copying: Vertex-copying growth, whose new nodes copy the edges of existing nodes and make a heavy-tailed degree.

gurus

• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.

gyre

• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.

gyroscope

• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.

haematopoiesis

• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.

hamiltonian

• Beam-beam map beam_beam_map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Cat map cat_map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.
• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
• Hénon-Heiles henon_heiles: The Hénon-Heiles system of a star in a galactic potential, whose motion is chaotic at higher energies.
• Nuclear quadrupole vibrations nuclear_quadrupole: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.
• Particle in the well k x^2 e^-x particle_in_a_well: A particle moving without friction in the potential U(x) = k x² e^-x.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Standard standard_map: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Two-body orbit two_body_orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.

hamiltonian path

• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.

handling time

• Mutualism mutualism: Two species that grow logistically and feed each other, above their carrying capacities.

harvesting

• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.
• Lotka-Volterra with harvesting lotka_volterra_with_harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.

health

• Obesity system obesity: The obesity system of adults who eat near home and work, one step per day.

heaps law

• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.

heat

• Cooling coffee cooling_coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).

heat bath

• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.
• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.

heavy tails

• Ornstein-Uhlenbeck with weakening restoring force weakening_ornstein_uhlenbeck: Brownian particles held by a restoring force that weakens with distance.

herding

• Kirman exchange rate with switching traders kirman_exchange_rate: Kirman's foreign exchange market, whose traders switch between fundamentalism and chartism like ants.
• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.
• Strömbom shepherding of a flock shepherding: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.

heterogeneity

• Annealed transient time motion annealed_transient_time_motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.
• Produce market produce_market: Hamill and Gilbert's produce market, where shoppers with lists of produce visit stalls at random.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.

heuristics

• Brock-Hommes heuristic switching brock_hommes: Brock and Hommes' asset market with traders switching between fundamentalists and trend chasers.

hexagons

• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.

hidden markov system

• Hidden Markov source hidden_markov_source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.

hierarchy

• Bonabeau hierarchies bonabeau_hierarchies: The Bonabeau system, where a hierarchy of strengths emerges above a critical density of agents.
• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.
• Hora and Tempus watchmakers watchmakers: Simon's fable of the watchmakers Hora and Tempus, told by Meadows, with and without stable subassemblies.

higher-order interactions

• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.

hill climbing

• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.

history dependence

• Ageing random walk ageing_random_walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.
• Pólya urn polya_urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.

hiv

• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.

homeostasis

• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.
• Polyhomeostatic neuron polyhomeostatic_neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.

hopf bifurcation

• Ikeda delay equation ikeda: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Stuart-Landau stuart_landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.

hormone

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.

hormone therapy

• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
• Prostate cancer under continuous androgen suppression continuous_androgen_suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.

hormones

• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.

host-parasitoid

• Nicholson-Bailey nicholson_bailey: Hosts and parasitoids in discrete generations, whose equilibrium is unstable to growing oscillations.

household

• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.

human mobility

• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.
• Exploration and preferential return exploration_preferential_return: The exploration and preferential return model of human mobility of Song and coauthors.

humanitarian logistics

• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.

hunter-gatherers

• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.

hybrid

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Circuits under periodic impulses impulsive_circuits: An RLC circuit driven by a periodic train of current impulses, L C x'' + R C x' + x = A Σ δ(t - k T).
• Lumped cardiovascular system cardiovascular: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.

hydrodynamics

• Brownian particle near a wall brownian_near_wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.
• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.
• Toner-Tu hydrodynamics toner_tu: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.

hydrophobicity

• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.

hyperbolic

• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.

hyperbolic growth

• Finite-time blow-up finite_time_blow_up: Hyperbolic growth x' = a x^p, which diverges at the finite time x0^(1 - p) / (a (p - 1)) for p > 1.

hyperchaos

• Hyperchaotic four-dimensional Rössler flow hyperchaotic_rossler: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
• Towel maps towel_map: Rössler's towel map, a three-dimensional map with two positive Lyapunov exponents.

hypercycle

• Hypercycles hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.

hypergraph

• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.
• Kuramoto oscillators with three-body coupling hypergraph_kuramoto: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Nonlinear three-body consensus on temporal hypergraphs hypergraph_consensus: The nonlinear three-body consensus system on a random hypergraph of triangles that switch on and off.
• Public goods game on hypergraphs hypergraph_public_goods: The public goods game on a hypergraph, played in every hyperedge, with Fermi imitation.
• SIS on hypergraphs with heterogeneous exposure and a nonlinear infection kernel hypergraph_exposure_sis: The SIS contagion of St-Onge et al. in the environments of a hypergraph, with a nonlinear kernel.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.
• Size-biased random walk on hypergraphs hypergraph_random_walk: The size-biased random walk on a hypergraph, p_j' = sum_i p_i T_ij - p_j, from one node.
• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.
• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.

hysteresis

• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.

ice ages

• Saltzman-Maasch glaciation saltzman_glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.

ice-albedo feedback

• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Zero-dimensional energy balance with ice albedo energy_balance: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.

imitation

• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.
• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

immune

• Immune competition immune_competition: The kinetic system of the competition between tumour cells and immune cells.

immunity

• Record-dynamics epidemic of resistant residents record_dynamics_epidemic: Yearly strains that infect every resident less resistant than their strength, who become immune.

immunosuppression

• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.

immunotherapy

• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Kirschner-Panetta system with TGF-beta and siRNA sirna_tgf_beta: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Oncolytic adenovirus with T cells and dendritic-cell injection oncolytic_virus_dendritic: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Predator-prey tumour-immune system of TCR gene therapy tcr_gene_therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.
• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.
• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.

impacts

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Bouncing ball bouncing_ball: A ball that falls under gravity and bounces off the floor, losing speed by a coefficient of restitution.
• Newton's cradle newtons_cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.

impulses

• Circuits under periodic impulses impulsive_circuits: An RLC circuit driven by a periodic train of current impulses, L C x'' + R C x' + x = A Σ δ(t - k T).
• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.

incoherent feedforward

• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.

income distribution

• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.

independent trials

• Bernoulli process bernoulli_process: Independent Bernoulli processes, one success or failure per step, whose success counts are binomial.

individual growth

• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.

inductive reasoning

• El Farol el_farol: Arthur's El Farol bar problem, whose attendance self-organizes around the capacity.
• Minority game and its evolutionary version minority_game: The minority game of an odd number of agents with two random strategies each.

inequality

• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

inertia

• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Inertial and overdamped Brownian motion brownian_motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.

infection

• Uninfected and infected tumour cells under an oncolytic virus oncolytic_virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.

infinite alphabet

• Power-series automaton power_series_automaton: The automaton τ(x)(n) = x(n) - t x(n + 1) over the ring K[[t]] of formal power series on a ring.
• Real quadratic automaton real_quadratic_automaton: The automaton τ(x)(n) = x(n + 1) - x(n)^2 over the reals on a ring.
• Self-indexed shift over a group alphabet self_indexed_shift: The map τ(x)(g) = x(g · x(g)) over the group as alphabet, continuous and equivariant but no automaton.
• Subtraction automaton over the naturals subtraction_automaton: The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.

influentials

• Belief diffusion with influentials influentials: The belief diffusion system with influentials who care passionately and weigh more with their neighbours.

information

• Polyhomeostatic neuron polyhomeostatic_neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.

information theory

• Hidden Markov source hidden_markov_source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.
• Markov source markov_source: A Markov source, a sequence of symbols in which each symbol depends on the one before it.

innovation

• Arthur-Polak technology evolution arthur_polak: Arthur and Polak's evolution of technology, which builds logic functions from NAND gates.
• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.
• Combinatorial creative phase transition creative_phase_transition: The diversity of a combinatorial system, which jumps from sparse to full above a critical rule density.
• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• NK technology system nk_technology: McNerney's system of technological improvement, whose cost falls as a power of the cumulative production.
• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.
• Threshold adoption threshold_adoption: The threshold adoption system in social circles, started by the 2.5% of agents with the lowest thresholds.

insider threat

• Insider sabotage insider_sabotage: The insider sabotage system of employees whose risk of sabotage builds up over the days of a year.

instability

• Car following with delay delayed_car_following: A chain of cars that react to the speed difference to the car ahead after a reaction time.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.

institution

• Fractional-reserve banking fractional_reserve_banking: Customers and banks whose lending of excess reserves multiplies the base money by 1 / ratio.

insulin

• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.

integers

• Unbounded-alphabet automaton unbounded_alphabet_automaton: An additive automaton over the nonnegative integers, x'(n) = x(n - 1) + x(n), from a single 1.

integrable

• Circular billiard circular_billiard: A particle moving freely inside a circle and reflecting specularly off its wall.

integral operator

• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.

integrate and fire

• Mirollo-Strogatz pulse-coupled oscillators mirollo_strogatz: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.

interaction matrix

• Generalized Lotka-Volterra microbial community generalized_lotka_volterra: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).

interdependent networks

• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.

intermediate growth

• Automata on the Grigorchuk group grigorchuk_automaton: The majority automaton on a ball of the Grigorchuk group, a group of intermediate growth.

intermittency

• Intermittency maps intermittency_map: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
• Record-dynamics epidemic of resistant residents record_dynamics_epidemic: Yearly strains that infect every resident less resistant than their strength, who become immune.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.

intermittent

• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.

international trade

• Exchange rates in international trade exchange_rates: Hamill and Gilbert's staged adjustment of trade and the exchange rate to a rise in home prices.

interruptions

• Hora and Tempus watchmakers watchmakers: Simon's fable of the watchmakers Hora and Tempus, told by Meadows, with and without stable subassemblies.

interval map

• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
• Random two-branch interval map random_branch_map: Orbits of a random map of [0, 1] with the two branches (1 + x) / 2 and mu x².

invariant density

• Bernoulli doubling map doubling_map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.
• Generalized tent map skew_tent_map: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.

invariants

• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.

inventory

• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.

inversion

• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.

invertible

• Involutive automaton involutive_automaton: A binary automaton on a ring that is its own inverse, with the minimal memory set {-1, 0, ..., n}.

involution

• Involutive automaton involutive_automaton: A binary automaton on a ring that is its own inverse, with the minimal memory set {-1, 0, ..., n}.

irreversibility

• Ehrenfest urn system of two boxes ehrenfest_urn: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.

isotope labelling

• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.

iterated

• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.

iterated function system

• Iterated function systems by the chaos game chaos_game: Points that draw the attractor of an iterated function system by the chaos game.
• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.
• Random two-branch interval map random_branch_map: Orbits of a random map of [0, 1] with the two branches (1 + x) / 2 and mu x².

jams

• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• Intelligent driver system intelligent_driver_system: The intelligent driver system of cars on a ring road.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Nagel-Schreckenberg nagel_schreckenberg: The Nagel-Schreckenberg traffic system on a ring road, where jams emerge above a critical density.
• Optimal velocity optimal_velocity: The optimal velocity system of cars on a ring road, where uniform flow breaks into stop-and-go waves.
• Ring-road driver agents ring_road_drivers: Drivers on a circular highway, where small random errors of speed set off traffic jams.

julia set

• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.

jump

• SIR sir: An epidemic that moves individuals from susceptible to infected to recovered.
• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS sis: An infection without immunity, where the recovered become susceptible again and the disease is endemic.
• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

jump process

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Binary ant decision binary_ant_decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Birth-death process and M/M/1 queue birth_death_queue: Independent M/M/1 queues, birth-death processes with constant arrival and service rates.
• Calm and nervous rats rats: A colony of calm and nervous rats, in which nervousness spreads by contact and fades on its own.
• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• Fully connected Ising system with Glauber dynamics curie_weiss: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.
• Generalized stochastic predator-prey stochastic_predator_prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.
• Genetic toggle switch toggle_switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Maki-Thompson rumours maki_thompson: The Maki-Thompson rumour system started by one spreader, which leaves about 20% of people ignorant.
• Michaelis-Menten as ODEs and as a jump process michaelis_menten: The enzyme reaction E + S <-> C -> E + P of Michaelis and Menten, as rate equations or as jumps.
• Negative autoregulation and bursty gene expression negative_autoregulation: A gene that represses its own transcription and makes its protein in bursts.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Neutral macroecology neutral_macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.
• Poisson process poisson_process: Independent Poisson processes, counters that grow by one at exponentially distributed waiting times.
• Pólya urn polya_urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.
• Random walks random_walk: Independent walkers on the integers that step at random times, a continuous-time random walk.
• Repressilator repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.
• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.
• TAP growth with random catalysis and autocatalytic sets tap_catalysis: The TAP growth of goods from combinations of goods, with random catalysis of their production functions.
• Two-phase zone exchange two_phase_zone_exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.
• Zero-range process zero_range_process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

jumps

• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.
• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.
• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.

kam

• Conservative Hénon map conservative_henon_map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.
• Standard standard_map: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.

kinetic

• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Criminality and security criminality: The kinetic system of criminality in a society and the security forces that act against it.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• Exchange with debt limit and with saving propensity kinetic_wealth_exchange: A kinetic wealth exchange system with a saving propensity or a debt limit.
• Immune competition immune_competition: The kinetic system of the competition between tumour cells and immune cells.
• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.
• Kinetic opinion formation kinetic_opinion: The kinetic system of opinion formation over a discrete grid of opinions.
• Kinetic wealth and welfare dynamics kinetic_wealth_welfare: The kinetic theory of wealth classes, with a welfare policy when the tax is positive.
• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.

kinetic theory

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.

kirchhoff

• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.

kleiber law

• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

knock-out

• Boolean system of the mammalian cell cycle mammalian_cell_cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.

kolkata paise restaurant

• Kolkata Paise Restaurant kolkata_restaurant: The Kolkata Paise Restaurant problem, where diners spread over restaurants that serve one meal each.

koopman

• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.

kronecker product

• Kronecker graphs kronecker_graph: The Kronecker graph of two graphs, whose adjacency matrix is the Kronecker product of theirs.

labour

• Car wash queue car_wash: Chesney's Clean and Wax hand car wash, a shadow account of the revenue of its workers' labour.
• Labour market labour_market: Hamill and Gilbert's labour market of firms posting wage offers and unemployed workers searching.

labour exploitation

• Ajani's Choice ajanis_choice: Chesney's Ajani's Choice, migrant farm workers and employers among whom underpayment emerges.

labour standards

• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.
• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.

lagrangian

• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.

lags

• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.

lamplighter

• Lamplighter lamplighter_automaton: The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.

land surface

• Coupled soil and deep temperature maps soil_temperature: The coupled maps of the soil surface temperature and the deeper soil temperature.

land use

• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Von Thünen land use von_thunen: A square plain around one market at its centre, whose every cell grows the crop of highest bid rent.

landscape

• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.

lanes

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Social force pedestrians social_force: Two crowds walking in opposite directions along a periodic corridor, which form lanes.

langevin

• Langevin car-following toward a density-dependent speed langevin_traffic: Cars on a ring road whose speeds relax with noise towards a desired speed set by the local density.

langevin equation

• Langevin equation with colored noise colored_noise_langevin: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.

language

• Miller's random typewriting random_typewriting: Miller's random typewriting, monkeys whose random keystrokes give Zipf's law of word frequencies.
• Naming game naming_game: The minimal naming game, where agents with empty inventories agree on one word.
• Simon-Yule growth simon_yule: The Simon system of a text that grows from one word, whose word counts follow a power law.

large deviations

• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.

lateral inhibition

• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.

lattice

• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Bak-Tang-Wiesenfeld sandpile sandpile: The Bak-Tang-Wiesenfeld sandpile on a lattice with open edges, started from random stable heights.
• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.
• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Cyclic cellular automaton cyclic_cellular_automaton: Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
• Diffusion automaton diffusion_automaton: A cellular automaton of diffusion, c' = c + r Σ (c_neighbour - c), started from a square of ones.
• Discrete master equation discrete_master_equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.
• Drossel-Schwabl forest fire forest_fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Elementary automata elementary_automaton: An elementary cellular automaton by its Wolfram number, started from a single live cell by default.
• Employment takeover employment_takeover: The employment takeover system on a periodic checkerboard, with half of each kind coloured at random.
• FHP lattice gas fhp_lattice_gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Fredkin's self-replicating parity automaton fredkin_replicator: Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.
• Game of Life game_of_life: Conway's Game of Life, or another life-like rule, on a torus.
• Group-valued difference automaton group_difference_automaton: The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.
• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.
• Highlife and the planar automaton zoo highlife: HighLife, B36/S23, or any other life-like automaton of the planar zoo by its rule, on a torus.
• Hodgepodge machine hodgepodge: The hodgepodge machine, a system of a chemical reaction whose waves form spirals.
• Involutive automaton involutive_automaton: A binary automaton on a ring that is its own inverse, with the minimal memory set {-1, 0, ..., n}.
• Ising ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.
• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Langton's ant langtons_ant: Langton's ant, which wanders chaotically for about 10,000 steps before it builds a periodic highway.
• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.
• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.
• Majority rule automaton majority_rule: Voters that take the majority opinion of their neighbourhood, which coarsens into domains.
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).
• Manna sandpile manna_sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Nagel-Schreckenberg nagel_schreckenberg: The Nagel-Schreckenberg traffic system on a ring road, where jams emerge above a critical density.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Nilpotent automaton of class n nilpotent_automaton: A binary automaton on a ring that is nilpotent of class n: n steps send every configuration to 0.
• Noisy automata noisy_automata: An elementary cellular automaton whose outputs are flipped at random with a small probability.
• Olami-Feder-Christensen earthquakes: The Olami-Feder-Christensen earthquake system with open edges, started from random forces.
• Potts potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• Power-series automaton power_series_automaton: The automaton τ(x)(n) = x(n) - t x(n + 1) over the ring K[[t]] of formal power series on a ring.
• Public goods game public_goods_game: The spatial public goods game in groups of five neighbours, with Fermi imitation.
• Q2R reversible Ising automaton q2r: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.
• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Real quadratic automaton real_quadratic_automaton: The automaton τ(x)(n) = x(n + 1) - x(n)^2 over the reals on a ring.
• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.
• Site percolation site_percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.
• Spatial prisoner's dilemma with best-neighbour and Fermi imitation spatial_prisoners_dilemma: The spatial prisoner's dilemma of Nowak and May, with best-neighbour or Fermi imitation.
• Subtraction automaton over the naturals subtraction_automaton: The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.
• Turing machine (three-state busy beaver) turing_machine: A Turing machine, a head with an internal state that reads, writes, and moves on a tape of symbols.
• Unbounded-alphabet automaton unbounded_alphabet_automaton: An additive automaton over the nonnegative integers, x'(n) = x(n - 1) + x(n), from a single 1.
• Von Thünen land use von_thunen: A square plain around one market at its centre, whose every cell grows the crop of highest bid rent.
• XY xy: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.
• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

lattice gas

• FHP lattice gas fhp_lattice_gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.

leadership

• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.

learning

• Adaptrode synapse adaptrode: An Adaptrode synapse driven by a train of input pulses that stops after a while.
• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.

learning curve

• NK technology system nk_technology: McNerney's system of technological improvement, whose cost falls as a power of the cumulative production.

least effort

• Mandelbrot least-effort word frequencies mandelbrot_least_effort: The word frequencies that minimise the mean cost of a language over its entropy, a power law of rank.

leukaemia

• Bone-marrow and blood cells with blasts and cancer stem cells bone_marrow_blood_leukaemia: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Red-cell production under 6-mercaptopurine erythropoiesis_mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.

levitation

• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.

life

• Game of Life game_of_life: Conway's Game of Life, or another life-like rule, on a torus.
• Game of Life on a rewired small-world network small_world_life: Conway's Game of Life on a small-world network rewired from the Moore lattice of a torus.
• Highlife and the planar automaton zoo highlife: HighLife, B36/S23, or any other life-like automaton of the planar zoo by its rule, on a torus.
• Life in a tree life_in_a_tree: The Game of Life on the free group F4, a tree of degree 8, surjective but not pre-injective.

life cycle

• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.

light

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.

limit cycle

• Brusselator ODE brusselator_ode: The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Fractional Van der Pol fractional_van_der_pol: The fractional Van der Pol oscillator D^alpha x = y, D^alpha y = mu (1 - x^2) y - x.
• Hopf hopf_form: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Saltzman-Maasch glaciation saltzman_glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Stuart-Landau stuart_landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Tunnel-diode oscillator tunnel_diode_oscillator: A tunnel-diode oscillator: an inductor and a capacitor in parallel with a negative-resistance diode.
• Van der Pol van_der_pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.
• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.

linear

• Fredkin's self-replicating parity automaton fredkin_replicator: Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
• Lamplighter lamplighter_automaton: The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
• Linear automata over rings linear_ring_automaton: The discrete Laplacian as a linear automaton on a ring, over the field of modulus elements.
• Surjective non-pre-injective linear automaton on F2 surjective_linear_automaton: A linear automaton over K^2 on the free group F2, surjective but not pre-injective.

linear dynamics

• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.

linear map

• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.

linear program

• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.
• Flux balance, flux variability, and parsimonious FBA flux_balance: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.
• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.

linear system

• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

linear-quadratic

• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.

linear-quadratic regulator

• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.

liquid crystal

• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.

livelihood

• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.

livestock

• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.

lockdown

• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.

log kill

• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Skipper-Schabel-Wilcox log-kill skipper_log_kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.

logical network

• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.

logistic

• Generalized logistic (Richards) growth generalized_logistic: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.

logistic map

• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.

lognormal

• Gibrat gibrat_process: Firms that grow by Gibrat's law of proportionate effect, which makes their sizes lognormal.

long memory

• Fractional Brownian motion fractional_brownian_motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.

lookup tables

• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.

lumped system

• Lumped cardiovascular system cardiovascular: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.

macroecology

• Neutral macroecology neutral_macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.

macroeconomics

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.

magnetism

• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.

majority

• Automata on the Grigorchuk group grigorchuk_automaton: The majority automaton on a ball of the Grigorchuk group, a group of intermediate growth.
• Automata on the free group free_group_automaton: The majority automaton on a ball of the free group, a tree in which every cell follows its neighbours.
• Baumslag-Solitar groups baumslag_solitar_automaton: The majority automaton on a ball of the Baumslag-Solitar group BS(m, n), of exponential growth.
• Galam majority galam_majority: The Galam majority system, where random groups adopt their majority until one opinion wins.
• Heisenberg heisenberg_automaton: The majority automaton on a ball of the discrete Heisenberg group, of polynomial growth of degree 4.
• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.

map

• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Coupled map lattices on networks coupled_map_lattices_on_networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Julia sets julia: The filled Julia set of z -> z^2 + c, the starting points whose orbit stays bounded (Dilão 2023 §5.5).
• Mandelbrot set mandelbrot: The Mandelbrot set, the points c whose orbit of z -> z^2 + c from 0 stays bounded (Dilão 2023 §5.5).
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.

market

• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.
• Cobweb cobweb: A market whose price converges to or oscillates around equilibrium as supply lags the price.
• Cournot duopoly with beliefs cournot_duopoly: Two Cournot duopolists that adjust their output to their beliefs about each other.
• Edgeworth and Red Cross barter barter: Barter of chocolate and cigarettes, in an Edgeworth box of two barterers or among many prisoners.
• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.
• Labour market labour_market: Hamill and Gilbert's labour market of firms posting wage offers and unemployed workers searching.
• Produce market produce_market: Hamill and Gilbert's produce market, where shoppers with lists of produce visit stalls at random.
• Riccati price adjustment riccati_prices: A price that moves with excess demand under linear demand and saturating quadratic supply.
• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.
• Zero-intelligence double auction double_auction: Gode and Sunder's double auction of zero-intelligence traders, a step per trading period.

market impact

• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.

market sight

• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

marketing

• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.

markov chain

• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.
• Ehrenfest urn system of two boxes ehrenfest_urn: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.
• Hidden Markov source hidden_markov_source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.
• Markov source markov_source: A Markov source, a sequence of symbols in which each symbol depends on the one before it.

mass action

• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.

mass conserving

• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.

mass extinctions

• Rothman excitable carbon cycle carbon_cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.

mass spring damper

• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.

master equation

• Discrete master equation discrete_master_equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.
• Size-biased random walk on hypergraphs hypergraph_random_walk: The size-biased random walk on a hypergraph, p_j' = sum_i p_i T_ij - p_j, from one node.

master stability function

• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.

matching

• Labour market labour_market: Hamill and Gilbert's labour market of firms posting wage offers and unemployed workers searching.
• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.

maxwell

• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Traders and exchange rates coupled through arbitrage as lattice electrodynamics gauge_arbitrage_market: Exchange-rate potentials on links as lattice Maxwell fields, sourced by traders who move a good on sites.

mean field

• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Fully connected Ising system with Glauber dynamics curie_weiss: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Kuramoto kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Landau mean-field relaxation landau_relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Ott-Antonsen reduction ott_antonsen: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.
• Theta neurons and QIF networks theta_neurons: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.

measurement error

• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.

mechanics

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Bouncing ball bouncing_ball: A ball that falls under gravity and bounces off the floor, losing speed by a coefficient of restitution.
• Circular billiard circular_billiard: A particle moving freely inside a circle and reflecting specularly off its wall.
• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Harmonic oscillator harmonic_oscillator: A particle on a spring, optionally damped.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• Magnetic pendulum magnetic_pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• N-body and solar system n_body: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Newton's cradle newtons_cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.
• Particle in the well k x^2 e^-x particle_in_a_well: A particle moving without friction in the potential U(x) = k x² e^-x.
• Pendulum pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Projectile projectile: A particle launched under uniform gravity, optionally slowed by linear drag.
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Rotating balls on rods rotating_balls_on_rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Two-body orbit two_body_orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.

mechatronics

• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.
• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.
• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.
• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.

memory

• Adaptrode synapse adaptrode: An Adaptrode synapse driven by a train of input pulses that stops after a while.
• Fractional Lorenz fractional_lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
• Fractional Van der Pol fractional_van_der_pol: The fractional Van der Pol oscillator D^alpha x = y, D^alpha y = mu (1 - x^2) y - x.
• Fractional damped oscillator fractional_oscillator: The fractional damped oscillator D^alpha x = v, D^alpha v = -omega^2 x - 2 gamma v.
• Fractional relaxation fractional_relaxation: Fractional relaxation D^alpha x = -lambda x, which decays as the Mittag-Leffler function E_alpha.
• Random walk with autoregressive velocity random_walk_with_memory: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.

memristor

• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Memristor oscillator memristor_oscillator: The simplest memristor oscillator: a capacitor, an inductor, and a memristor in series.

merging

• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.

mesh

• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.

metabolic oscillations

• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.

metabolic scaling

• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

metabolism

• COMETS: dynamic flux balance on a lattice with diffusion comets: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.
• Dynamic flux balance analysis of a batch culture dynamic_fba: A batch culture of the E. coli core system, which needs the extra metabolism.
• Dynamic multi-species metabolic system dynamic_multispecies_fba: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.
• Flux balance, flux variability, and parsimonious FBA flux_balance: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.
• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.

metapopulation

• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.

metastability

• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.

metastasis

• Primary and secondary sites with a migration delay under chemotherapy delayed_metastasis: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.

microbial community

• Generalized Lotka-Volterra microbial community generalized_lotka_volterra: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).

microphase separation

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.

microswimmers

• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.

milling

• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.

min proteins

• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.

minimal

• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.

minority game

• Kolkata Paise Restaurant kolkata_restaurant: The Kolkata Paise Restaurant problem, where diners spread over restaurants that serve one meal each.
• Minority game and its evolutionary version minority_game: The minority game of an odd number of agents with two random strategies each.

mixed phase space

• Dynamical billiards of Sinai, mushroom, and open three-disk shapes billiards: A particle moving freely on a billiard table and reflecting specularly off its walls.

mixed-mode oscillations

• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.

mixing

• Baker baker_map: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
• Cat map cat_map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.

mode locking

• Circle circle_map: The sine circle map θ' = θ + Ω - K / (2π) sin 2πθ (mod 1), which locks into Arnold tongues.

model predictive control

• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.

modern slavery

• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.

molecular computation

• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.

molecular dynamics

• Lennard-Jones fluid lennard_jones_fluid: Lennard-Jones particles in a periodic square box, optionally held at a temperature by a thermostat.

momentum

• Ballistic pendulum ballistic_pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Newton's cradle newtons_cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.

monetary policy

• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

money

• Fractional-reserve banking fractional_reserve_banking: Customers and banks whose lending of excess reserves multiplies the base money by 1 / ratio.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.
• Stock-flow consistent monetary economy stock_flow_consistent_economy: A stock-flow consistent economy, a closed circuit of money between households, firms, and a bank.
• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

monte carlo

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.
• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.
• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.
• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.
• Lattice Maxwell and gauge link fields lattice_gauge_fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.

morse-smale

• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.

mortality

• Agent-based SIR with mortality, waning immunity, incubation, and lockdown agent_based_sir: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.

motifs

• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.

multi-valued

• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.

multilayer

• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.

multilevel selection

• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.

multiplex

• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.

multiplicative noise

• Brownian particle near a wall brownian_near_wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.
• Multiplicative noise conventions multiplicative_noise: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.

multiplicative process

• Gibrat gibrat_process: Firms that grow by Gibrat's law of proportionate effect, which makes their sizes lognormal.
• Kesten processes kesten_process: Firms whose sizes follow Kesten's process, a random multiplicative factor plus a random positive term.
• Multiplicative multiplicative_process: Levy and Solomon's multiplicative process, whose floor at a fraction of the mean makes a power-law tail.

multiscale

• Multiscale tumour angiogenesis tumour_angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.

multistability

• Magnetic pendulum magnetic_pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.

mutation

• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.
• Iterated prisoner's dilemma on a lattice with strategies of cooperation rounds iterated_lattice_dilemma: Argun's evolutionary iterated prisoner's dilemma on a lattice, with strategies of n cooperation rounds.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.

mutualism

• Mutualism mutualism: Two species that grow logistically and feed each other, above their carrying capacities.

myelination

• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.

nadh

• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.

nanomedicine

• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.

nanotubes

• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.

naturals

• Subtraction automaton over the naturals subtraction_automaton: The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.

necrosis

• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.

negative feedback

• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.
• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.
• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.
• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.

neighborhood

• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks gregoire_chate: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Vicsek system with metric, topological, vision-cone, and delayed alignment vicsek: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

neighbours

• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.

nematic

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Ginelli nematic particles ginelli_nematics: Polar particles with nematic alignment, which order along a common axis but move both ways along it.

net energy

• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.

network

• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Barabási-Albert growth, with Bianconi-Barabási fitness barabasi_albert: Barabási-Albert growth, whose degrees follow a power law P(k) ~ k^-3 by preferential attachment.
• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.
• Bouchaud-Mézard wealth bouchaud_mezard_wealth: Bouchaud and Mezard's wealth dynamics, random multiplicative returns with exchange between neighbours.
• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
• Centralities, assortativity, communities, and motifs network_structure_analysis: A network recorded with its centralities, degree mixing, communities, and triangle z-score.
• Cognitive consistency agents cognitive_consistency: Agents whose attitudes spread by conversation and settle in a belief network that seeks consistency.
• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.
• Coupled map lattices on networks coupled_map_lattices_on_networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.
• DebtRank cascades debtrank: The DebtRank cascade of distress from a set of banks through their interbank loans.
• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Diffusion network_diffusion: Diffusion on a network, du_i/dt = D Σ_j A_ij (u_j - u_i), which spreads a unit mass to every node.
• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.
• Erdos-Renyi and Watts-Strogatz graphs random_graph_ensembles: The construction of an Erdős-Rényi or a Watts-Strogatz random graph, one step of new edges at a time.
• Evolutionary dynamics with game transitions game_transitions: Evolutionary dynamics with game transitions on a random regular graph, after Su, McAvoy, Wang, and Nowak.
• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.
• Firm and bank credit networks credit_networks: A credit market where firms sample banks and switch to cheaper lenders, and defaults hit the lenders.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Game of Life on a rewired small-world network small_world_life: Conway's Game of Life on a small-world network rewired from the Moore lattice of a torus.
• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.
• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.
• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.
• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.
• Ising on a graph graph_ising: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Kinetic evacuation of a network of rooms evacuation: The kinetic system of the evacuation of a building whose rooms form a network.
• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.
• Kronecker graphs kronecker_graph: The Kronecker graph of two graphs, whose adjacency matrix is the Kronecker product of theirs.
• Kuramoto on networks kuramoto_on_networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Linear network herding network_herding: Agents whose linear choices herd on the past choices of the agents they are linked to.
• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.
• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.
• PageRank iterations on networks pagerank: The PageRank iteration r_i' = (1 - d) / n + d Σ_j A_ij r_j / k_j on an undirected network.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.
• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.
• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.
• Random walks network_random_walk: Random walkers on a network, which visit every node of a connected graph in proportion to its degree.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• SIR on a network as an agent system network_sir: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.
• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.
• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.
• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.
• Sznajd sznajd: The Sznajd system, where agreeing pairs of agents convince their neighbours.
• Theta neurons and QIF networks theta_neurons: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.
• Three-state opinion with undecided agents three_state_opinion: The three-state opinion system of agents against, undecided, and for, on a random regular network.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.
• Vertex-copying network growth vertex_copying: Vertex-copying growth, whose new nodes copy the edges of existing nodes and make a heavy-tailed degree.
• Voter system voter_system: A voter system with one voter per node of a network.
• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.
• Watts threshold cascades watts_cascades: The Watts threshold system of global cascades started by a few random seeds.

network effect

• Fixed-line phone adoption phone_adoption: Hamill and Gilbert's adoption of fixed-line phones by British households from 1951.

network geometry

• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.

network motifs

• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.

networks

• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.

neural field

• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.

neural mass

• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.
• Wilson-Cowan wilson_cowan: The Wilson-Cowan system of interacting excitatory and inhibitory populations of neurons.

neural network

• Asymmetric diluted Hopfield network diluted_hopfield: The asymmetric diluted Hopfield network, which recalls a stored pattern below its memory capacity.
• Echo state network echo_state_network: An echo state network, a reservoir of leaky tanh units with random recurrent weights driven by a sine.
• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.
• Fukai-Tanaka winner-take-all network winners_take_all: A winner-take-all network of units with self-limited growth and mutual inhibition.
• Hopfield network hopfield: A Hopfield network that stores patterns with Hebb's rule and recalls the first from a corrupted copy.
• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.

neuron

• FitzHugh-Nagumo ODE fitzhugh_nagumo_ode: The FitzHugh-Nagumo neuron, the two-variable reduction of Hodgkin and Huxley's excitable membrane.
• Hodgkin-Huxley hodgkin_huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.
• Integrate-and-fire integrate_and_fire: The leaky integrate-and-fire neuron, tau v' = -(v - rest) + R I, which spikes and resets at a threshold.
• Izhikevich izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Polyhomeostatic neuron polyhomeostatic_neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
• Theta neurons and QIF networks theta_neurons: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.

neurons

• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.

neutral networks

• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.

neutral theory

• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.
• Neutral birth-death with hopping between boxes neutral_clustering: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Neutral macroecology neutral_macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.

next generation

• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.
• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.

niche theory

• Consumer-resource models of substitutable and essential resources consumer_resource: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.

nilpotent

• Heisenberg heisenberg_automaton: The majority automaton on a ball of the discrete Heisenberg group, of polynomial growth of degree 4.
• Nilpotent automaton of class n nilpotent_automaton: A binary automaton on a ring that is nilpotent of class n: n steps send every configuration to 0.

node turnover

• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.

noise

• Chemical Langevin equation of the Schlögl reaction chemical_langevin_equation: The chemical Langevin equation of Schlögl's bistable reaction, whose noise switches between two states.
• Chemical Langevin equation of the enzyme network chemical_langevin_network: The chemical Langevin equation of the enzyme reaction E + S <-> C -> E + P.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.
• Langevin equation with colored noise colored_noise_langevin: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.
• Multiplicative noise conventions multiplicative_noise: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.
• Negative autoregulation and bursty gene expression negative_autoregulation: A gene that represses its own transcription and makes its protein in bursts.
• Noisy automata noisy_automata: An elementary cellular automaton whose outputs are flipped at random with a small probability.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.
• Stochastic resonance stochastic_resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.

noise traders

• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.

non-closed image

• Subtraction automaton over the naturals subtraction_automaton: The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.

non-ergodic

• Scaled Brownian motion scaled_brownian_motion: Scaled Brownian motion, dx = sqrt(2 K alpha (t + t0)^(alpha - 1)) dW, with <x^2> ~ 2 K t^alpha.

non-growing network

• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.
• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.

non-local

• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.

nonequilibrium

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.
• Zero-range process zero_range_process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

nonequilibrium steady state

• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Overdamped particle in a tilted periodic potential tilted_periodic_potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.

nonlocal coupling

• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.

nonreciprocal

• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.
• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.

nonrenewable resource

• Oil economy oil_economy: Meadows' oil economy, capital that grows on a nonrenewable resource and collapses as it runs out.

nonstationary

• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.

normal form

• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• Hopf hopf_form: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
• Saddle-node saddle_node_form: The saddle-node normal form x' = μ - x², whose two equilibria ±√μ meet and vanish at μ = 0.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Takens-Bogdanov takens_bogdanov_form: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.

normal modes

• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Standing waves on a string standing_waves: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.

not sofic

• Context-free subshift context_free_subshift: A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.

notch-delta

• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.

nuclear physics

• Nuclear quadrupole vibrations nuclear_quadrupole: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.

nutrient

• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.

obesity

• Obesity system obesity: The obesity system of adults who eat near home and work, one step per day.

ode

• Acquired and induced resistance with phenotype switching drug_resistance_switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.
• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.
• Adler equation adler_equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Assembly growth along the assembly index assembly_growth: The numbers of objects at successive assembly indices, each discovered from the level below.
• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.
• Bathtub bathtub: A bathtub, one stock of water with an inflow from the faucet and an outflow through the drain.
• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• Bistability on a catalyst pellet catalyst_pellet: The surface concentration of a catalyst pellet, bistable through Langmuir-Hinshelwood kinetics.
• Blood glucose stabilised by antagonistic insulin and glucagon loops glucose_dual_hormone_control: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.
• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.
• Brusselator ODE brusselator_ode: The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
• Capital stock capital_stock: Meadows' capital stock, grown by reinvested output and drained by depreciation.
• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Cascade classifier cascade_classifier: A cascade classifier, a column of stages that separates a narrow size class of particles by an air flow.
• Cell-cycle compartments with killing, blocking, and recruiting agents cell_cycle_chemotherapy: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Chua double and multiple scrolls chua: Chua's circuit, whose double-scroll or multiscroll attractor is chaotic.
• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.
• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Competition competition: Two species that compete for the same resources, which coexist or exclude each other.
• Consumer-resource models of substitutable and essential resources consumer_resource: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.
• Cooling coffee cooling_coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).
• Core and periphery vasculature under anti-angiogenic therapy vascular_core_periphery: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Coupled Colpitts oscillators coupled_colpitts: Two detuned chaotic Colpitts oscillators coupled through the difference of their second variables.
• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.
• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Cubic autocatalysis cubic_autocatalysis: Cubic autocatalysis A + 2B -> 3B with the decay B -> C in a continuously fed reactor.
• Cucker-Smale flocking cucker_smale_flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Cumulative drug toxicity drug_toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• Cusp normal forms cusp_form: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
• DC motor with an inertia dc_motor: A DC motor whose armature circuit of components drives an inertia on a Lagrangian coordinate.
• Daisyworld daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.
• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.
• Diffusion network_diffusion: Diffusion on a network, du_i/dt = D Σ_j A_ij (u_j - u_i), which spreads a unit mass to every node.
• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.
• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.
• Duffing duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.
• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.
• Exponential growth exponential_growth: A population that grows or declines at a constant per-capita rate.
• Exponential-linear tumour growth exponential_linear_growth: A tumour that grows exponentially while small and linearly once its growth reaches a limiting rate.
• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Faraday-disk charger faraday_disk: A battery charger driven by a Faraday disk generator whose own current strengthens its field.
• Finite-time blow-up finite_time_blow_up: Hyperbolic growth x' = a x^p, which diverges at the finite time x0^(1 - p) / (a (p - 1)) for p > 1.
• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.
• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Food webs food_webs: A food web where basal species grow logistically and consumers live on Lotka-Volterra predation.
• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Generalized Lotka-Volterra microbial community generalized_lotka_volterra: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).
• Generalized logistic (Richards) growth generalized_logistic: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.
• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.
• Gissinger geodynamo geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.
• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Gompertz growth gompertz: A tumour whose growth slows as the logarithm of the room left, dn/dt = r n log(K / n).
• Gompertz growth with a vascular carrying capacity dynamic_carrying_capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.
• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Harmonic oscillator harmonic_oscillator: A particle on a spring, optionally damped.
• Harvesting farmers harvesting: A socio-ecological system of farmer agents harvesting a logistically regrowing resource.
• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.
• Hopf hopf_form: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Hyperchaotic four-dimensional Rössler flow hyperchaotic_rossler: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
• Hypercycles hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.
• Hénon-Heiles henon_heiles: The Hénon-Heiles system of a star in a galactic potential, whose motion is chaotic at higher energies.
• Ideal gas ideal_gas: An ideal gas of non-interacting molecules in a reflecting box.
• Intelligent driver system intelligent_driver_system: The intelligent driver system of cars on a ring road.
• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.
• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• Kirschner-Panetta system with TGF-beta and siRNA sirna_tgf_beta: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Landau mean-field relaxation landau_relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Langford attractor langford: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
• Lennard-Jones fluid lennard_jones_fluid: Lennard-Jones particles in a periodic square box, optionally held at a temperature by a thermostat.
• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.
• Logistic growth logistic_growth: A population that grows logistically towards its carrying capacity.
• Logistic growth with a varying capacity varying_capacity_logistic: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.
• Lorenz lorenz: The Lorenz-63 system of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
• Lorenz-84 lorenz_84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.
• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.
• Lotka-Volterra lotka_volterra: Prey and predators whose populations oscillate through predation.
• Lotka-Volterra with harvesting lotka_volterra_with_harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Magnetic levitation under alternating impulses magnetic_levitation: A conductive ball levitated in a vacuum by a field that pushes and pulls it with alternating impulses.
• Magnetic pendulum magnetic_pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.
• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Memristor oscillator memristor_oscillator: The simplest memristor oscillator: a capacitor, an inductor, and a memristor in series.
• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Michaelis-Menten as ODEs and as a jump process michaelis_menten: The enzyme reaction E + S <-> C -> E + P of Michaelis and Menten, as rate equations or as jumps.
• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.
• Mutualism mutualism: Two species that grow logistically and feed each other, above their carrying capacities.
• N-body and solar system n_body: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.
• Normal and tumour competition under radiation radiotherapy_competition: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.
• Normal, tumour, and immune cells under chemotherapy de_pillis_radunskaya: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Norton-Simon Gompertzian cell kill norton_simon: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.
• Nuclear quadrupole vibrations nuclear_quadrupole: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.
• Oil economy oil_economy: Meadows' oil economy, capital that grows on a nonrenewable resource and collapses as it runs out.
• Oncolytic adenovirus with T cells and dendritic-cell injection oncolytic_virus_dendritic: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Optimal chemotherapy dosing optimal_chemotherapy: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.
• Optimal velocity optimal_velocity: The optimal velocity system of cars on a ring road, where uniform flow breaks into stop-and-go waves.
• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.
• Ott-Antonsen reduction ott_antonsen: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.
• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.
• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.
• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.
• Particle in the well k x^2 e^-x particle_in_a_well: A particle moving without friction in the potential U(x) = k x² e^-x.
• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.
• Pendulum pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.
• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.
• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.
• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Power-law tumour growth power_law_growth: A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.
• Predator-prey tumour-immune system of TCR gene therapy tcr_gene_therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.
• Projectile projectile: A particle launched under uniform gravity, optionally slowed by linear drag.
• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
• Prostate cancer under continuous androgen suppression continuous_androgen_suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.
• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.
• Pulsed electrochemical reactor pulsed_electrochemical_reactor: A stirred electrochemical reactor whose electrode is driven by a train of current pulses.
• Quadratic system with a slow manifold and a finite Koopman embedding koopman_slow_manifold: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.
• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.
• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.
• RLC circuit rlc_circuit: A series RLC circuit written as Kirchhoff's voltage law, L q'' + R q' + q / C = V cos(ω t).
• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.
• Regenerative braking regenerative_braking: A car braked by a Faraday disk generator on its wheel, which charges a supercapacitor.
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Renewable fishery with capital fishery_capital: Meadows' fishing fleet, capital that grows on a renewable resource and can overshoot it.
• Replicator dynamics and hawks and doves replicator_dynamics: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.
• Resistance subpopulations under three drugs multidrug_resistance: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.
• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Riccati price adjustment riccati_prices: A price that moves with excess demand under linear demand and saturating quadratic supply.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.
• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.
• Rotating balls on rods rotating_balls_on_rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.
• Rothman excitable carbon cycle carbon_cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.
• Rössler rossler: The Rössler system, a chaotic flow with one folded band for its standard parameters.
• Rössler-driven Lorenz and generalized synchronization rossler_driven_lorenz: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.
• SIR sir: An epidemic that moves individuals from susceptible to infected to recovered.
• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.
• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS sis: An infection without immunity, where the recovered become susceptible again and the disease is endemic.
• Saddle-node saddle_node_form: The saddle-node normal form x' = μ - x², whose two equilibria ±√μ meet and vanish at μ = 0.
• Saltzman-Maasch glaciation saltzman_glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Satellite transfer satellite_transfer: A satellite moved between Kepler orbits of equal angular momentum by radial thrust.
• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Size-biased random walk on hypergraphs hypergraph_random_walk: The size-biased random walk on a hypergraph, p_j' = sum_i p_i T_ij - p_j, from one node.
• Skipper-Schabel-Wilcox log-kill skipper_log_kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.
• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.
• Solow solow: An economy whose capital per worker approaches its steady state through saving and depreciation.
• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.
• Sprott jerk sprott_jerk: Sprott's jerk equation x''' = -A x'' - x' + |x| - 1, chaotic near A = 0.6.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.
• Stommel thermohaline boxes stommel: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.
• Stuart-Landau stuart_landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Subcritical Hopf normal form subcritical_hopf_form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.
• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.
• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.
• T cells, tumour, and anti-PD-1 checkpoint inhibitor anti_pd1_checkpoint: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Takens-Bogdanov takens_bogdanov_form: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.
• Thermostat thermostat: A room heated by a furnace under a thermostat while heat leaks to the outside.
• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.
• Thomas cyclically symmetric attractor thomas: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.
• Three-enzyme adaptation circuits of Ma et al. enzyme_adaptation_circuit: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.
• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.
• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.
• Three-way fate decision on the compactified elliptic umbilic three_way_fate_decision: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.
• Tumour and vascular volume under an angiogenesis inhibitor hahnfeldt: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta tumour_immune_suppression: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.
• Tunnel-diode oscillator tunnel_diode_oscillator: A tunnel-diode oscillator: an inductor and a capacitor in parallel with a negative-resistance diode.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.
• Two-body orbit two_body_orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.
• Two-compartment pharmacokinetics two_compartment_pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.
• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.
• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.
• Uninfected and infected tumour cells under an oncolytic virus oncolytic_virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.
• Van der Pol van_der_pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.
• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.
• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.
• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.
• Von Bertalanffy growth von_bertalanffy: A tumour that grows by the balance of anabolism and catabolism, dn/dt = r n^p - d n.
• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.
• Yeast glycolysis oscillations in a flow reactor glycolysis_oscillations: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.
• Zero-dimensional energy balance with ice albedo energy_balance: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.
• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

odometer

• Adding machine (odometer) adding_machine: The adding machine r → r + 1 on the q-adic integers, the odometer acting on the q-ary tree.

oligopoly

• Cournot duopoly with beliefs cournot_duopoly: Two Cournot duopolists that adjust their output to their beliefs about each other.
• Static oligopoly equilibria oligopoly_equilibria: The equilibria of a Cournot duopoly and of a pricing game of firms with loyal customers and switchers.

oncolytic virus

• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.

open system

• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.

open-ended evolution

• Assembly growth along the assembly index assembly_growth: The numbers of objects at successive assembly indices, each discovered from the level below.

opinion

• Belief diffusion with influentials influentials: The belief diffusion system with influentials who care passionately and weigh more with their neighbours.
• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.
• Cognitive consistency agents cognitive_consistency: Agents whose attitudes spread by conversation and settle in a belief network that seeks consistency.
• Deffuant bounded confidence deffuant: The Deffuant bounded confidence system, whose opinions settle into about 1 / (2 * confidence) clusters.
• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.
• Galam majority galam_majority: The Galam majority system, where random groups adopt their majority until one opinion wins.
• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.
• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
• Kinetic opinion formation kinetic_opinion: The kinetic system of opinion formation over a discrete grid of opinions.
• Majority rule automaton majority_rule: Voters that take the majority opinion of their neighbourhood, which coarsens into domains.
• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.
• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Naming game naming_game: The minimal naming game, where agents with empty inventories agree on one word.
• Nonlinear three-body consensus on temporal hypergraphs hypergraph_consensus: The nonlinear three-body consensus system on a random hypergraph of triangles that switch on and off.
• Social impact theory social_impact: The social impact system on a full grid, with random persuasiveness and supportiveness.
• Sznajd sznajd: The Sznajd system, where agreeing pairs of agents convince their neighbours.
• Three-state opinion with undecided agents three_state_opinion: The three-state opinion system of agents against, undecided, and for, on a random regular network.
• Threshold adoption threshold_adoption: The threshold adoption system in social circles, started by the 2.5% of agents with the lowest thresholds.
• Voter system voter_system: A voter system with one voter per node of a network.
• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.

opinions

• Gauge theory of agents on a network gauge_network_agents: Agent phases on the nodes of a network that relax towards the transported phases of their neighbours.

optical forces

• Diffusion in a speckle field diffusion_in_a_speckle_field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.

optical tweezers

• Hydrodynamically coupled particles in blinking traps hydrodynamically_coupled_traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.
• Optical trap including oscillating and blinking traps optical_trap: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.

optimal control

• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Optimal chemotherapy dosing optimal_chemotherapy: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.
• Pontryagin capital and consumption optimal_consumption: The Ramsey economy that chooses the consumption path maximising its discounted utility.
• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.

optimal growth

• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.

optimisation

• Simulated annealing, genetic algorithms, and differential evolution evolutionary_optimisers: A population of candidates that searches a two-dimensional landscape, one generation per step.

orbits

• Satellite transfer satellite_transfer: A satellite moved between Kepler orbits of equal angular momentum by radial thrust.

order book

• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.

organizations

• Insider sabotage insider_sabotage: The insider sabotage system of employees whose risk of sabotage builds up over the days of a year.

origin of life

• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.

oscillation

• Adler equation adler_equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Agent-based Lotka-Volterra agent_based_lotka_volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• Brusselator ODE brusselator_ode: The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• ENSO delay oscillator enso: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Harmonic oscillator harmonic_oscillator: A particle on a spring, optionally damped.
• Hutchinson hutchinson: Hutchinson's delayed logistic equation, which oscillates around K when r * tau exceeds π/2.
• Ikeda delay equation ikeda: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
• Lotka-Volterra lotka_volterra: Prey and predators whose populations oscillate through predation.
• Lotka-Volterra with harvesting lotka_volterra_with_harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Nicholson-Bailey nicholson_bailey: Hosts and parasitoids in discrete generations, whose equilibrium is unstable to growing oscillations.
• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.
• Pendulum pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Stuart-Landau stuart_landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.
• Tunnel-diode oscillator tunnel_diode_oscillator: A tunnel-diode oscillator: an inductor and a capacitor in parallel with a negative-resistance diode.
• Two-mass spring-damper between walls two_mass_spring_damper: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.
• Van der Pol van_der_pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.

oscillations

• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Drosophila circadian clock circadian_clock: Goldbeter's system of the Drosophila circadian clock, PER that represses its own transcription.
• Effector cells, tumour, and IL-2 under adoptive therapy kirschner_panetta: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Repressilator repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.
• Wilson-Cowan wilson_cowan: The Wilson-Cowan system of interacting excitatory and inhibitory populations of neurons.

oscillator

• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Fractional damped oscillator fractional_oscillator: The fractional damped oscillator D^alpha x = v, D^alpha v = -omega^2 x - 2 gamma v.
• Goodwin oscillator goodwin_oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Linear-quadratic regulated oscillator lqr_oscillator: A harmonic oscillator that a linear-quadratic regulator brings to rest.
• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.

oscillator death

• Winfree system winfree: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

oscillators

• Delayed Kuramoto delayed_kuramoto: Kuramoto phase oscillators coupled all to all through signals that arrive after a delay.
• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Kuramoto kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Kuramoto on networks kuramoto_on_networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Kuramoto oscillators with three-body coupling hypergraph_kuramoto: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Kuramoto-Sakaguchi chimera kuramoto_sakaguchi_chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.
• Winfree system winfree: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

outbreak

• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

overgrazing

• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.

overlapping portfolios

• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.

overload

• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.

pagerank

• PageRank iterations on networks pagerank: The PageRank iteration r_i' = (1 - d) / n + d Σ_j A_ij r_j / k_j on an undirected network.

pair interaction

• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Cucker-Smale flocking cucker_smale_flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Motility-induced phase separation motility_induced_phase_separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.

pair potential

• Lennard-Jones fluid lennard_jones_fluid: Lennard-Jones particles in a periodic square box, optionally held at a temperature by a thermostat.

paleoclimate

• Saltzman-Maasch glaciation saltzman_glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.

panic

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.

paradox of enrichment

• Generalized stochastic predator-prey stochastic_predator_prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.

parametric resonance

• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.

pareto

• Bouchaud-Mézard wealth bouchaud_mezard_wealth: Bouchaud and Mezard's wealth dynamics, random multiplicative returns with exchange between neighbours.

parity

• Fredkin's self-replicating parity automaton fredkin_replicator: Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.

particles

• Bird swarm with pair potentials bird_swarm: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Social force pedestrians social_force: Two crowds walking in opposite directions along a periodic corridor, which form lanes.

partnerships

• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.
• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.

pascal triangle

• Unbounded-alphabet automaton unbounded_alphabet_automaton: An additive automaton over the nonnegative integers, x'(n) = x(n - 1) + x(n), from a single 1.

pattern formation

• Bristle patterning by lateral inhibition on a ring bristle_lateral_inhibition: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.
• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.

payments

• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.

pde

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.
• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Decaying turbulence decaying_turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.
• Discrete-velocity Boltzmann gas of Carleman discrete_velocity_boltzmann: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• Fisher-KPP fisher_kpp: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).
• FitzHugh-Nagumo spirals fitzhugh_nagumo: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Gray-Scott gray_scott: The Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.
• Heat diffusion heat_diffusion: Heat diffusing on a plate from a hot square in its middle, du/dt = D laplacian(u).
• Heat on an L-shaped domain heat_l_shape: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.
• Hyperbolic chemotaxis of cells and a chemical hyperbolic_chemotaxis: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Keller-Segel chemotaxis keller_segel: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.
• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.
• Random-walk traders with arbitrage arbitrage_traders: Schwichtenberg's diffusion of a good by random-walk traders, minimally coupled to exchange rates.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Schnakenberg schnakenberg: The Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.
• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.
• Standing waves on a string standing_waves: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.
• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.
• Toner-Tu hydrodynamics toner_tu: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.
• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.
• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

pedestrians

• Bellomo's kinetic crowd in a street with walls and panic kinetic_crowd: Bellomo's kinetic crowd of two groups walking in opposite directions along a street with walls.
• Social force pedestrians social_force: Two crowds walking in opposite directions along a periodic corridor, which form lanes.

pendulum

• Pendulum swing-up pendulum_swing_up: A pendulum swung up to its inverted position by a bang-bang acceleration of its pivot.

perceptron

• Binary perceptron and random K-SAT perceptron_satisfiability: A binary perceptron that must store random patterns, revealed one per step, as cardinality constraints.

percolation

• Cascades of failures in interdependent networks interdependent_network_cascade: The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
• Continuum percolation continuum_percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Explosive percolation by the Achlioptas product rule explosive_percolation: Explosive percolation, where the Achlioptas product rule delays and sharpens the giant component.
• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Site percolation site_percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.
• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.

period doubling

• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
• Logistic map logistic_map: The logistic map x_{t+1} = r x_t (1 - x_t), which reaches chaos through period doubling as r grows.
• Ricker map of overcompensating generations ricker: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.

periodic

• Logistic growth with a varying capacity varying_capacity_logistic: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.

persistence

• Persistent random walk persistent_random_walk: Walkers on the integers that keep the direction of their last step with a probability.

petri net

• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.

pharmacokinetics

• Gompertzian cell-cycle phases under tamoxifen tamoxifen_cell_cycle: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Two-compartment pharmacokinetics two_compartment_pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.

pharmacology

• Cumulative drug toxicity drug_toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• Two-compartment pharmacokinetics two_compartment_pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.

phase

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Adler equation adler_equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Kuramoto kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Kuramoto on networks kuramoto_on_networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Kuramoto oscillators with three-body coupling hypergraph_kuramoto: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Kuramoto-Sakaguchi chimera kuramoto_sakaguchi_chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.
• Winfree system winfree: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

phase ordering

• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.

phase oscillators

• Phase units with activity-dependent conduction delays adaptive_delay_network: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.
• Theta neurons and QIF networks theta_neurons: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.
• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.

phase separation

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Motility-induced phase separation motility_induced_phase_separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.

phase space

• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Bertin-Droz-Grégoire kinetic flocking bertin_droz_gregoire: The kinetic theory of flocking, whose polar order grows above a transition density.
• Boltzmann boltzmann_equation: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Enskog equations enskog: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.

phase transition

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.
• Bonabeau hierarchies bonabeau_hierarchies: The Bonabeau system, where a hierarchy of strengths emerges above a critical density of agents.
• Calm and nervous rats rats: A colony of calm and nervous rats, in which nervousness spreads by contact and fades on its own.
• Combinatorial creative phase transition creative_phase_transition: The diversity of a combinatorial system, which jumps from sparse to full above a critical rule density.
• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.
• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• Fully connected Ising system with Glauber dynamics curie_weiss: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Fully connected branching and annihilation absorbing_birth_death: The fully connected branching and annihilation process, active above kappa = nu until it dies out.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Ising ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Ising on a graph graph_ising: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Landau mean-field relaxation landau_relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Potts potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• Site percolation site_percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.

phenotype switching

• Acquired and induced resistance with phenotype switching drug_resistance_switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.

phoresis

• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.

phosphorylation

• Three-node phosphorylation oscillator phosphorylation_oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.

physiology

• Mackey-Glass mackey_glass: The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.

phytoplankton

• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.

pid

• PID governance loop pid_loop: A work process whose output a PID controller keeps at a set point against a disturbance.
• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.

piecewise

• Exponential-linear tumour growth exponential_linear_growth: A tumour that grows exponentially while small and linearly once its growth reaches a limiting rate.

piecewise linear

• Generalized tent map skew_tent_map: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.
• Lozi map lozi_map: The Lozi map x' = 1 + y - a |x|, y' = b x, the piecewise linear Hénon map with a strange attractor.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Tent tent_map: The tent map x' = μ min(x, 1 - x), chaotic with Lyapunov exponent ln μ for μ above 1.

piezoelectric

• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.

plasma

• Vlasov vlasov_equation: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.

plasticity

• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.

polar

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.

polarization

• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.
• Kinetic opinion formation kinetic_opinion: The kinetic system of opinion formation over a discrete grid of opinions.

policy

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.

policy resistance

• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.

pollution

• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.

polymer

• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.

polymers

• Undirected and directed polymer assembly pool polymer_assembly_pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.

pontryagin

• Minimum-time control of a free particle, a harmonic oscillator, and a saddle minimum_time_control: A linear plant steered to the origin of its phase space in minimum time by a sampled bang-bang control.
• Optimal adoptive therapy and IL-2 dosing optimal_immunotherapy: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Optimal chemotherapy dosing optimal_chemotherapy: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.

popularity-similarity

• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.

population

• Exponential growth exponential_growth: A population that grows or declines at a constant per-capita rate.
• Fibonacci rabbits fibonacci_rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.
• Hutchinson hutchinson: Hutchinson's delayed logistic equation, which oscillates around K when r * tau exceeds π/2.
• Logistic growth logistic_growth: A population that grows logistically towards its carrying capacity.
• Logistic growth with a varying capacity varying_capacity_logistic: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.
• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Ricker map of overcompensating generations ricker: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.

population genetics

• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Moran process moran_process: The Moran process, a population of constant size where fitter traits fix by selection and drift.
• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.

population growth

• Droop cell-quota growth droop_cell_quota: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.

positive feedback

• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.

potential

• Particle in the well k x^2 e^-x particle_in_a_well: A particle moving without friction in the potential U(x) = k x² e^-x.

power grid

• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.

power law

• Bak-Tang-Wiesenfeld sandpile sandpile: The Bak-Tang-Wiesenfeld sandpile on a lattice with open edges, started from random stable heights.
• Barabási priority queues priority_queues: The Barabási priority queue, whose waiting times follow a power law as the choice becomes greedy.
• Cascading sample-space-reducing process cascading_sample_space_reducing: Cascading sample-space-reducing processes, whose visits follow the power law i^(-mu) for any mu.
• Drossel-Schwabl forest fire forest_fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• Exponential growth observed at exponential random times exponential_growth_at_random_times: Exponential growth observed at exponentially distributed times, whose observed values follow a power law.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Generalized mass action and S-system power-law systems biochemical_systems_theory: A biochemical system of power-law rates, in the S-system or the generalized mass action form.
• Kesten processes kesten_process: Firms whose sizes follow Kesten's process, a random multiplicative factor plus a random positive term.
• Mandelbrot least-effort word frequencies mandelbrot_least_effort: The word frequencies that minimise the mean cost of a language over its entropy, a power law of rank.
• Manna sandpile manna_sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• Miller's random typewriting random_typewriting: Miller's random typewriting, monkeys whose random keystrokes give Zipf's law of word frequencies.
• Multiplicative multiplicative_process: Levy and Solomon's multiplicative process, whose floor at a fraction of the mean makes a power-law tail.
• Olami-Feder-Christensen earthquakes: The Olami-Feder-Christensen earthquake system with open edges, started from random forces.
• Ornstein-Uhlenbeck with weakening restoring force weakening_ornstein_uhlenbeck: Brownian particles held by a restoring force that weakens with distance.
• Power-law tumour growth power_law_growth: A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.
• Sample-space-reducing process sample_space_reducing: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.
• Simon-Yule growth simon_yule: The Simon system of a text that grows from one word, whose word counts follow a power law.
• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

power series

• Power-series automaton power_series_automaton: The automaton τ(x)(n) = x(n) - t x(n + 1) over the ring K[[t]] of formal power series on a ring.

pre-injective

• Real quadratic automaton real_quadratic_automaton: The automaton τ(x)(n) = x(n + 1) - x(n)^2 over the reals on a ring.

prebiotic

• Hypercycles hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.

precession

• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.

predator-prey

• Agent-based Lotka-Volterra agent_based_lotka_volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• Food webs food_webs: A food web where basal species grow logistically and consumers live on Lotka-Volterra predation.
• Generalized stochastic predator-prey stochastic_predator_prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.
• Lotka-Volterra lotka_volterra: Prey and predators whose populations oscillate through predation.
• Lotka-Volterra with harvesting lotka_volterra_with_harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Predator-prey tumour-immune system of TCR gene therapy tcr_gene_therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.
• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.
• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.
• Wolf and sheep wolf_sheep: The wolf and sheep predation system with one grass patch per cell.

preferential attachment

• Barabási-Albert growth, with Bianconi-Barabási fitness barabasi_albert: Barabási-Albert growth, whose degrees follow a power law P(k) ~ k^-3 by preferential attachment.
• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.
• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.

preferential return

• Exploration and preferential return exploration_preferential_return: The exploration and preferential return model of human mobility of Song and coauthors.

price convergence

• Bilateral barter of two or three goods goods_traders: Traders who wander a grid and barter two or three goods with their neighbours.

price discovery

• Gooding double auction with adaptive quotes gooding_double_auction: Gooding's double auction, whose buyers and sellers move their quotes towards trade and then apart.

price elasticity

• Heterogeneous household demand by utility and by elasticities household_demand: Hamill and Gilbert's heterogeneous households, whose demand for food traces a curve as its price rises.

prices

• Toy Trader and Three-Goods Trader with money toy_trader: Gooding's Toy Trader, traders with money and toys who buy the cheapest toy they can see.

probability

• Discrete master equation discrete_master_equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.

productivity shocks

• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.

prostate cancer

• Intermittent androgen suppression switched by PSA thresholds ideta_intermittent_androgen: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Piecewise-linear intermittent androgen suppression piecewise_linear_ias: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Prostate cancer driven by activated androgen receptors prostate_androgen_receptor: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
• Prostate cancer under continuous androgen suppression continuous_androgen_suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.
• Prostate cancer with an androgen cell quota and PSA prostate_cell_quota: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.

protein

• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.
• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.

protein corona

• Protein corona binding on a nanoparticle nanoparticle_corona: The corona that albumin, high-density lipoprotein, and fibrinogen form on a nanoparticle in plasma.

protein interaction network

• Steady-state duplication-deletion network duplication_deletion: The steady-state duplication-deletion network of Jensen, whose connectance settles at Eq. 8.84.

public goods

• Public goods game public_goods_game: The spatial public goods game in groups of five neighbours, with Fermi imitation.
• Public goods game on hypergraphs hypergraph_public_goods: The public goods game on a hypergraph, played in every hyperedge, with Fermi imitation.

pullers

• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.

pulse coupling

• Mirollo-Strogatz pulse-coupled oscillators mirollo_strogatz: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.

pulses

• Pulsed electrochemical reactor pulsed_electrochemical_reactor: A stirred electrochemical reactor whose electrode is driven by a train of current pulses.
• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.

punctuated equilibria

• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.

punctuated equilibrium

• Competitive Jain-Krishna model with signed couplings competitive_jain_krishna: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Tangled Nature tangled_nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.

punishment

• Finite two-level birth-death process of defectors and punishers two_level_birth_death: The finite two-level birth-death process of defectors and altruistic punishers in competing groups.
• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.
• Multilevel selection with non-additive pairwise group conflict nonlinear_pairwise_group_conflict: The density of groups of defectors and altruistic punishers under pairwise group conflict.
• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.

purchasing power

• Three-sector purchasing power decoupled by feedback categorized_purchasing_power: Forrest's categorized purchasing power of money, three sectors that feedback can steer one by one.

pushers

• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.

q-adic integers

• Adding machine (odometer) adding_machine: The adding machine r → r + 1 on the q-adic integers, the odometer acting on the q-ary tree.

qcd

• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.

quantum

• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.

quasi-geostrophic

• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.

quasispecies

• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.

quenched mean field

• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.

queue

• Car wash queue car_wash: Chesney's Clean and Wax hand car wash, a shadow account of the revenue of its workers' labour.

queueing

• Birth-death process and M/M/1 queue birth_death_queue: Independent M/M/1 queues, birth-death processes with constant arrival and service rates.

queues

• Barabási priority queues priority_queues: The Barabási priority queue, whose waiting times follow a power law as the choice becomes greedy.
• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.

radiotherapy

• Linear-quadratic cell survival and biologically effective dose lq_cell_survival: The linear-quadratic survival of cells after a total dose, with its biologically effective dose.
• Linear-quadratic impulsive irradiation with a radiosensitizer lq_radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Normal and tumour competition under radiation radiotherapy_competition: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.

railways

• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.

ramachandran

• Early-stage protein folding intermediate early_stage_folding_intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.

ramp

• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.

random

• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.

random community

• Random Lotka-Volterra with higher-order interactions higher_order_random_lotka_volterra: A random Lotka-Volterra community with third-order interactions.

random energy system

• Adaptive climbing in finite populations adaptive_climbing: A finite population that climbs a random fitness landscape by mutation and strong selection.

random graph

• Configuration model with stub matching and rewiring configuration_model: The configuration model, the maximally random network of a given degree sequence.
• Erdos-Renyi and Watts-Strogatz graphs random_graph_ensembles: The construction of an Erdős-Rényi or a Watts-Strogatz random graph, one step of new edges at a time.
• Kronecker graphs kronecker_graph: The Kronecker graph of two graphs, whose adjacency matrix is the Kronecker product of theirs.
• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.

random map

• Random complex square-root function system inverse_julia: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.
• Random two-branch interval map random_branch_map: Orbits of a random map of [0, 1] with the two branches (1 + x) / 2 and mu x².

random observation

• Exponential growth observed at exponential random times exponential_growth_at_random_times: Exponential growth observed at exponentially distributed times, whose observed values follow a power law.

random walk

• Ageing random walk ageing_random_walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.
• Continuous-time random walk continuous_time_random_walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Cubic chaotic map and its chaotic walk cubic_chaotic_map: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
• Diffusion-limited aggregation diffusion_limited_aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Lévy walk levy_walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).
• Ornstein-Uhlenbeck with weakening restoring force weakening_ornstein_uhlenbeck: Brownian particles held by a restoring force that weakens with distance.
• Persistent random walk persistent_random_walk: Walkers on the integers that keep the direction of their last step with a probability.
• Random walk with autoregressive velocity random_walk_with_memory: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.
• Random walks network_random_walk: Random walkers on a network, which visit every node of a connected graph in proportion to its degree.
• Random walks random_walk: Independent walkers on the integers that step at random times, a continuous-time random walk.
• Random-walker epidemics on a grid and on a small world random_walker_epidemics: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Size-biased random walk on hypergraphs hypergraph_random_walk: The size-biased random walk on a hypergraph, p_j' = sum_i p_i T_ij - p_j, from one node.
• Superdiffusive Levy flight levy_flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

ratchet

• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.

rate-induced tipping

• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.

rates

• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.

reaction-diffusion

• Brusselator fields brusselator: The Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).
• Fisher-KPP fisher_kpp: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).
• Gray-Scott gray_scott: The Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.
• Metapopulation reaction-diffusion metapopulation: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Schnakenberg schnakenberg: The Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.

reactor

• Cubic autocatalysis cubic_autocatalysis: Cubic autocatalysis A + 2B -> 3B with the decay B -> C in a continuously fed reactor.

reals

• Real quadratic automaton real_quadratic_automaton: The automaton τ(x)(n) = x(n + 1) - x(n)^2 over the reals on a ring.

record dynamics

• Record-dynamics epidemic of resistant residents record_dynamics_epidemic: Yearly strains that infect every resident less resistant than their strength, who become immune.

recurrent network

• Elman recurrent network trained by backpropagation elman_network: Elman's simple recurrent network, which learns to predict the next bit of a temporal XOR stream.

red noise

• Hasselmann stochastic climate hasselmann: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.

region layers

• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.

regulation

• Daisyworld daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.
• Tragedy of the meadow commons tragedy_of_the_commons: Herders whose cows graze a shared meadow, each adding cows while a cow pays, until the grass collapses.

regulatory T cells

• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.

regulatory flux balance

• Regulatory flux balance analysis regulatory_fba: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.

reinforcement

• Pólya urn polya_urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.

reinforcement learning

• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.

relaxation oscillation

• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.

relaxation oscillations

• FitzHugh-Nagumo ODE fitzhugh_nagumo_ode: The FitzHugh-Nagumo neuron, the two-variable reduction of Hodgkin and Huxley's excitable membrane.

relaxation oscillator

• Van der Pol ice ages van_der_pol_ice_ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.

remodelling

• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.

renewable resource

• Renewable fishery with capital fishery_capital: Meadows' fishing fleet, capital that grows on a renewable resource and can overshoot it.

renewal

• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.
• Exponential growth observed at exponential random times exponential_growth_at_random_times: Exponential growth observed at exponentially distributed times, whose observed values follow a power law.

renewal process

• Annealed transient time motion annealed_transient_time_motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Bouchaud trap system trap_system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.
• Continuous-time random walk continuous_time_random_walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Lévy walk levy_walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).

replicator

• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Fredkin's self-replicating parity automaton fredkin_replicator: Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
• Highlife and the planar automaton zoo highlife: HighLife, B36/S23, or any other life-like automaton of the planar zoo by its rule, on a torus.
• Replicator dynamics and hawks and doves replicator_dynamics: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.

representative agent

• Brock-Mirman stochastic growth brock_mirman_growth: The Brock-Mirman economy, the optimal growth of a representative agent whose productivity is Markov.

repressilator

• Boolean repressilator boolean_repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.

reservoir computing

• Echo state network echo_state_network: An echo state network, a reservoir of leaky tanh units with random recurrent weights driven by a sine.
• Next-generation reservoir computer next_generation_reservoir: A next-generation reservoir computer whose variables follow the ridge readout of their monomials.

resistance

• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity angiogenesis_immunotherapy: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Two tumour clones under chemotherapy and dendritic-cell vaccine clonal_chemoimmunotherapy: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.

resonance

• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Stochastic resonance stochastic_resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.

resource allocation

• Kolkata Paise Restaurant kolkata_restaurant: The Kolkata Paise Restaurant problem, where diners spread over restaurants that serve one meal each.

retail

• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.
• Shops and digital markets shops: Shops on a grid that compete for the shoppers on every cell, in person or in a digital market.

reversals

• Gissinger geodynamo geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.

reverse bifurcations

• Cascais-Dilão-Noronha Gaussian map cdn_map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.

reversible

• HPP lattice gas hpp_lattice_gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.
• Q2R reversible Ising automaton q2r: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.

rewiring

• Adaptive SIS with rewiring adaptive_sis: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.

rewriting system

• Cantor cantor_set: The Cantor set as the substitution of every interval by its two outer parts.
• Koch koch_curve: The Koch curve as the L-system F → F+F--F+F drawn by a turtle.
• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.
• Sierpiński substitutions sierpinski_substitution: The Sierpiński triangle or carpet as the substitution of a polygon by scaled copies of itself.

riccati

• Riccati price adjustment riccati_prices: A price that moves with excess demand under linear demand and saturating quadratic supply.

ridesharing

• Ridesharing ridesharing: Riders who request trips on a city grid and cabs that a stable matching sends to them.

ring

• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.

rna

• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.

robot

• RobbyGA can collector and evolution by circumstance robby_ga: Mitchell's genetic algorithm that evolves strategies for Robby, a robot that collects cans in a room.

robotics

• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Light-sensitive robots with sensory delay light_seeking_robots: Light-sensitive robots in a circular well, whose sensory delay decides whether they gather in the light.
• Swarmtaxis swarmtaxis: The swarmtaxis algorithm, which moves a swarm of robots towards a beacon that only some of them see.

robustness

• Genotype spaces and neutral networks of RNA folds rna_neutral_networks: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.
• Motter-Lai cascades of betweenness overload motter_lai_overload: The cascade of overload failures of Motter and Lai after one node is removed.
• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.

rolls

• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.

rotating frame

• Restricted three-body problem restricted_three_body: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.

rotation

• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Rotating balls on rods rotating_balls_on_rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.

rotational diffusion

• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.

rouse chain

• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.

ruggedness

• NK fitness landscapes nk_landscape: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.

rumours

• Maki-Thompson rumours maki_thompson: The Maki-Thompson rumour system started by one spreader, which leaves about 20% of people ignorant.

sample space reduction

• Cascading sample-space-reducing process cascading_sample_space_reducing: Cascading sample-space-reducing processes, whose visits follow the power law i^(-mu) for any mu.
• Fragmentation fragmentation: The random fragmentation of an object of integer size, whose fragments keep breaking into smaller ones.
• Sample-space-reducing process sample_space_reducing: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.

sandpile

• Fixed-energy sandpile fixed_energy_sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• Zhang sandpile zhang_sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

satisfiability

• Binary perceptron and random K-SAT perceptron_satisfiability: A binary perceptron that must store random patterns, revealed one per step, as cardinality constraints.

scale-free

• Barabási-Albert growth, with Bianconi-Barabási fitness barabasi_albert: Barabási-Albert growth, whose degrees follow a power law P(k) ~ k^-3 by preferential attachment.
• Degree-class SIR on uncorrelated networks heterogeneous_mean_field_sir: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.
• Heterogeneous mean-field SIS heterogeneous_mean_field_sis: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Node-merging network of Kim et al. node_merging: The node-merging network of Kim et al., whose merging of linked nodes builds hubs without growth.
• Popularity-similarity network popularity_similarity_network: Popularity-similarity growth, scale-free and clustered, whose nodes attach by hyperbolic distance.
• Random failures and targeted attacks on networks network_attack: Failures or attacks on a scale-free network, robust to random failures but fragile to attacks on hubs.
• Vertex-copying network growth vertex_copying: Vertex-copying growth, whose new nodes copy the edges of existing nodes and make a heavy-tailed degree.

scales

• Container model of human mobility container_mobility: The container model of human mobility of Alessandretti, Aslak, and Lehmann, trajectories across scales.

schooling

• Couzin shoal with repulsion, alignment, and informed leaders couzin_leadership: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.

sde

• Active Brownian particles active_brownian_particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Active Brownian particles with local alignment aligning_active_brownian_particles: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Active Ornstein-Uhlenbeck particles active_ornstein_uhlenbeck_particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.
• Annealed transient time motion annealed_transient_time_motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Bouchaud-Mézard wealth bouchaud_mezard_wealth: Bouchaud and Mezard's wealth dynamics, random multiplicative returns with exchange between neighbours.
• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Brownian particle near a wall brownian_near_wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Chemical Langevin equation of the Schlögl reaction chemical_langevin_equation: The chemical Langevin equation of Schlögl's bistable reaction, whose noise switches between two states.
• Chemical Langevin equation of the enzyme network chemical_langevin_network: The chemical Langevin equation of the enzyme reaction E + S <-> C -> E + P.
• Chiral active particles chiral_active_particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.
• Correlated geometric Brownian motions correlated_geometric_brownian_motion: Two assets whose prices follow geometric Brownian motions driven by correlated Wiener processes.
• Depot-pumped Brownian particles depot_brownian_particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).
• Diffusion in a speckle field diffusion_in_a_speckle_field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.
• Fishery with measurement errors fishery_measurement_errors: Independent fisheries whose quota is set from a measurement of the stock with a Gaussian error.
• Flocking with active and passive agents active_passive_flocking: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Fractional Brownian motion fractional_brownian_motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.
• Geometric Brownian motion geometric_brownian_motion: An asset whose price follows geometric Brownian motion.
• Hasselmann stochastic climate hasselmann: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.
• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Inertial and overdamped Brownian motion brownian_motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.
• Kimura diffusion of allele frequencies kimura_diffusion: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Kramers escape kramers_escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.
• Langevin car-following toward a density-dependent speed langevin_traffic: Cars on a ring road whose speeds relax with noise towards a desired speed set by the local density.
• Langevin equation with colored noise colored_noise_langevin: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.
• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Light-sensitive robots with sensory delay light_seeking_robots: Light-sensitive robots in a circular well, whose sensory delay decides whether they gather in the light.
• Living crystals living_crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Motility-induced phase separation motility_induced_phase_separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Multiplicative noise conventions multiplicative_noise: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.
• On-off flashing ratchet flashing_ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Optical trap including oscillating and blinking traps optical_trap: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.
• Ornstein-Uhlenbeck with weakening restoring force weakening_ornstein_uhlenbeck: Brownian particles held by a restoring force that weakens with distance.
• Overdamped particle in a tilted periodic potential tilted_periodic_potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.
• Rigid Brownian and active dimers rigid_dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.
• Rössler with an Ornstein-Uhlenbeck control parameter ou_driven_rossler: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
• Scaled Brownian motion scaled_brownian_motion: Scaled Brownian motion, dx = sqrt(2 K alpha (t + t0)^(alpha - 1)) dW, with <x^2> ~ 2 K t^alpha.
• Shaken frictional spring-block chain shaken_spring_block: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.
• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.
• Stochastic on-off intermittency on_off_intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.
• Stochastic resonance stochastic_resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.
• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.

second order

• Bending of light by a mass light_bending: A light ray that passes a static mass and is deflected towards it.
• Coupled pendulums coupled_pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Double pendulum double_pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Driven damped oscillator driven_damped_oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Duffing duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• FLRW cosmology from the Einstein equations flrw_cosmology: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Fibonacci rabbits fibonacci_rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.
• Forced Toda oscillator forced_toda: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Frame dragging around a spinning mass kerr_frame_dragging: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Friedmann expansion friedmann_expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Gyroscope in a gimbal gimballed_gyroscope: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• Multiplier-accelerator multiplier_accelerator: Samuelson's multiplier-accelerator, Y_{t+1} = c (1 + v) Y_t - c v Y_{t-1} + G.
• Oppenheimer-Snyder collapse of a dust ball oppenheimer_snyder_collapse: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.
• Pendulum pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Pendulum on a sliding cart cart_pendulum: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Pendulum with a vertically or horizontally oscillating pivot parametric_pendulum: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Perihelion precession in the Schwarzschild spacetime schwarzschild_precession: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.
• RLC circuit rlc_circuit: A series RLC circuit written as Kirchhoff's voltage law, L q'' + R q' + q / C = V cos(ω t).
• Relativistic orbit equation relativistic_orbit: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Rotating swing rotating_swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.
• Sitnikov problem sitnikov: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Strongly nonlinear oscillator under periodic impulses impulsive_oscillator: An oscillator kicked by a periodic train of impulses of alternating sign.
• Störmer charge in a dipole field stormer_problem: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Van der Pol van_der_pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.

security

• Criminality and security criminality: The kinetic system of criminality in a society and the security forces that act against it.

segmentation

• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.

segregation

• Schelling schelling: A Schelling segregation system on a grid with at most one resident per cell.
• Stochastic Schelling stochastic_schelling: The stochastic Schelling system of agents moving between blocks, which segregates above beta = 2.

seir

• Watts-May SEIR with lasting partnerships watts_may_concurrency: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.

selection

• Assembly growth along the assembly index assembly_growth: The numbers of objects at successive assembly indices, each discovered from the level below.
• Eigen quasispecies eigen_quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.
• Undirected and directed polymer assembly pool polymer_assembly_pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.

self-organization

• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions boids: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Social force pedestrians social_force: Two crowds walking in opposite directions along a periodic corridor, which form lanes.

self-organized criticality

• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Co-evolving combinatorial critical model ccc_model: The co-evolving combinatorial critical model, whose diversity shows punctuated equilibria.
• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.

self-propulsion

• Active Brownian particles active_brownian_particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Active Ornstein-Uhlenbeck particles active_ornstein_uhlenbeck_particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Active polymers active_polymer: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.
• Bird swarm with pair potentials bird_swarm: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Chiral active particles chiral_active_particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.
• Depot-pumped Brownian particles depot_brownian_particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).
• Inertial active Brownian particles inertial_active_brownian_particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Run-and-tumble run_and_tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.

self-replication

• Gray-Scott gray_scott: The Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.

self-similar group

• Adding machine (odometer) adding_machine: The adding machine r → r + 1 on the q-adic integers, the odometer acting on the q-ary tree.

sentiment

• The Kauai population system on region layers kauai_population: One agent per resident of the census regions of a map, moving between home, work, and favourite places.

separation

• Cascade classifier cascade_classifier: A cascade classifier, a column of stages that separates a narrow size class of particles by an air flow.

sepsis

• Two-layer adaptive oscillator model of sepsis two_layer_sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.

set point

• Steerable receptor-effector negative feedback loop steerable_feedback_loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.

shepherding

• Strömbom shepherding of a flock shepherding: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.

shocks

• BGK bgk_equation: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Burgers burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.

shortest path

• Ant colony optimization on a road network ant_colony_optimization: Ants that find the shortest path between two points of a Delaunay road network, round after round.

sigmoid

• Gompertz growth gompertz: A tumour whose growth slows as the logarithm of the room left, dn/dt = r n log(K / n).
• Gompertz growth with a vascular carrying capacity dynamic_carrying_capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.

signalling

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation hormone_cascade_amplifier: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.

simplex

• Multilevel selection of cooperators, defectors, and punishers on the simplex trimorphic_multilevel_selection: The density of groups of cooperators, defectors, and altruistic punishers on the 2-simplex.

simplicial complex

• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Growing simplicial complex by triangle attachment growing_simplicial_complex: A simplicial complex that grows by gluing one triangle per step along an edge drawn uniformly.
• Simplicial contagion simplicial_contagion: The simplicial contagion of Iacopini et al., an SIS process on the links and triangles of a complex.
• Two-player games on links and three-body games on 2-simplices simplicial_social_dilemmas: The social dilemmas of Guo et al. on the links and the 2-simplices of a simplicial complex.

singular arc

• Optimal harvesting optimal_harvesting: A logistic resource harvested with the effort that maximises its discounted profit.

singularity

• Finite-time blow-up finite_time_blow_up: Hyperbolic growth x' = a x^p, which diverges at the finite time x0^(1 - p) / (a (p - 1)) for p > 1.

sir

• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.

sis

• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.

size structure

• Size-structured forest with shading and seed dispersal forest_size_structure: The McKendrick-von Foerster density of trees over position across a strip and height.

slow manifold

• Quadratic system with a slow manifold and a finite Koopman embedding koopman_slow_manifold: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.

slow-fast

• Memristive mixed-mode oscillator memristive_mixed_mode_oscillator: A slow-fast circuit with a cubic flux-controlled memristor, whose oscillations mix two amplitudes.
• Modified Chua with mixed-mode oscillations mixed_mode_chua: A modified Chua circuit on three time scales, whose oscillations mix small and large amplitudes.
• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.

small world

• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.
• Erdos-Renyi and Watts-Strogatz graphs random_graph_ensembles: The construction of an Erdős-Rényi or a Watts-Strogatz random graph, one step of new edges at a time.
• Game of Life on a rewired small-world network small_world_life: Conway's Game of Life on a small-world network rewired from the Moore lattice of a torus.

snowball

• Budyko-Sellers latitude system budyko_sellers: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.

social dilemmas

• Two-player games on links and three-body games on 2-simplices simplicial_social_dilemmas: The social dilemmas of Guo et al. on the links and the 2-simplices of a simplicial complex.

social dynamics

• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.

social influence

• Axelrod culture dissemination axelrod: Axelrod's culture dissemination system on a full grid with von Neumann neighbourhoods.
• Fair-trade food choice with peer influence fair_food: The choice between fair-trade and regular food of consumers who influence their neighbours.

socio-ecological

• Harvesting farmers harvesting: A socio-ecological system of farmer agents harvesting a logistically regrowing resource.

sofic

• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.

soliton

• Lenia and its expanded universe lenia: Lenia, a continuous automaton whose ring-kernel convolutions grow gliding, lifelike solitons.

solvable

• Baumslag-Solitar groups baumslag_solitar_automaton: The majority automaton on a ball of the Baumslag-Solitar group BS(m, n), of exponential growth.

spatial network

• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.

spatiotemporal chaos

• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Lorenz-96 lorenz_96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.

spde

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Langevin field theory of absorbing-state transitions with a conserved energy field directed_percolation_langevin: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.

speciation

• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.

species abundance

• Neutral Moran system with speciation neutral_biodiversity: A neutral community of constant size, where species arise by speciation and vanish by drift.
• Neutral macroecology neutral_macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.

spectral

• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Decaying turbulence decaying_turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.
• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Rayleigh-Bénard rayleigh_benard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.
• Swift-Hohenberg swift_hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.

spectral radius

• Quenched mean-field SIS quenched_mean_field_sis: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.

spiking

• Hodgkin-Huxley hodgkin_huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.
• Integrate-and-fire integrate_and_fire: The leaky integrate-and-fire neuron, tau v' = -(v - rest) + R I, which spikes and resets at a threshold.
• Izhikevich izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.
• Networks of leaky integrate-and-fire neurons with delays delayed_lif_network: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.

spin glass

• Asymmetric diluted Hopfield network diluted_hopfield: The asymmetric diluted Hopfield network, which recalls a stored pattern below its memory capacity.
• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Evolutionary Sherrington-Kirkpatrick game evolutionary_sk_game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• SK game of reinforcement learners sk_game: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.

spin-orbit coupling

• Keplerian dumbbell and spin-orbit coupling keplerian_dumbbell: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.

spins

• Edwards-Anderson spin glass edwards_anderson: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Fully connected Ising system with Glauber dynamics curie_weiss: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Ising ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Ising on a graph graph_ising: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Join-a-club spins join_a_club: Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
• Mattis spin glass mattis_spin_glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Potts potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• Q2R reversible Ising automaton q2r: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.
• Quantum Heisenberg system quantum_heisenberg: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Sherrington-Kirkpatrick spin glass sherrington_kirkpatrick: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.
• XY xy: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.

spirals

• Aliev-Panfilov cardiac tissue aliev_panfilov: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Barkley excitable medium with spirals and breakup barkley: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Complex Ginzburg-Landau complex_ginzburg_landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Cyclic cellular automaton cyclic_cellular_automaton: Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
• FitzHugh-Nagumo spirals fitzhugh_nagumo: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• Hodgepodge machine hodgepodge: The hodgepodge machine, a system of a chemical reaction whose waves form spirals.
• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.

springs

• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.

spurious drift

• Multiplicative noise conventions multiplicative_noise: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.

squirmers

• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.

stability

• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network nanotube_gap_junction_exchange: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.

stable intermediate forms

• Hora and Tempus watchmakers watchmakers: Simon's fable of the watchmakers Hora and Tempus, told by Meadows, with and without stable subassemblies.

standardization

• Complex agents with internal states and standardization standardized_agents: Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.

static

• Carbon-13 metabolic flux analysis carbon_13_mfa: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.
• Continuum percolation continuum_percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Elementary flux modes and extreme pathways elementary_flux_modes: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.
• Flux balance, flux variability, and parsimonious FBA flux_balance: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Gravity interaction system of urban scaling gravity_urban_scaling: The residents of a city at uniform positions in a disc, each interacting with the others as r^-gamma.
• Joint, community, and SteadyCom flux balance community_fba: Two members of the E. coli core system that share a medium, which needs the extra metabolism.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals metabolic_perturbations: The E. coli core system with some genes knocked out, which needs the extra metabolism.
• Linear-quadratic cell survival and biologically effective dose lq_cell_survival: The linear-quadratic survival of cells after a total dose, with its biologically effective dose.
• Mandelbrot least-effort word frequencies mandelbrot_least_effort: The word frequencies that minimise the mean cost of a language over its entropy, a power law of rank.
• Random energy system random_energy_system: The random energy system, 2^N quenched Gaussian energies of variance N / 2 at a temperature.
• Static oligopoly equilibria oligopoly_equilibria: The equilibria of a Cournot duopoly and of a pricing game of firms with loyal customers and switchers.
• Von Thünen land use von_thunen: A square plain around one market at its centre, whose every cell grows the crop of highest bid rent.
• West-Brown-Enquist distribution network and Kleiber scaling west_brown_enquist: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

stationary distribution

• Random two-branch interval map random_branch_map: Orbits of a random map of [0, 1] with the two branches (1 + x) / 2 and mu x².

stdp

• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.

stem cells

• Bone-marrow and blood cells with blasts and cancer stem cells bone_marrow_blood_leukaemia: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.
• Normal and leukaemic stem cells with feedback regulation leukaemia_stem_cells: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and malignant stem-cell compartments with feedback and therapy stem_cell_differentiation: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.

stick-slip

• Burridge-Knopoff spring-block fault burridge_knopoff: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Mass-spring oscillator with static and Coulomb friction friction_oscillator: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.

stiff

• Kuramoto-Sivashinsky kuramoto_sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Oregonator oregonator: The scaled Oregonator, the reduction of the FKN mechanism of the Belousov-Zhabotinsky reaction.

stigmergy

• Ant colony optimization on a road network ant_colony_optimization: Ants that find the shortest path between two points of a Delaunay road network, round after round.
• Binary ant decision binary_ant_decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Pheromone foraging pheromone_foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Stigmergic predator-prey automaton stigmergic_predator_prey: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.

stochastic

• Adleman DNA computation of a Hamiltonian path dna_computation: Adleman's DNA computation of a Hamiltonian path, by random ligation of edge oligos in a test tube.
• Bak-Sneppen bak_sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• Contact process contact_process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Drossel-Schwabl forest fire forest_fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• FHP lattice gas fhp_lattice_gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• Forest fire on hexagonal lattices and graphs hexagonal_forest_fire: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Intrinsic-instability populations intrinsic_instability: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
• Kinouchi-Copelli excitable network kinouchi_copelli: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.
• Maki-Thompson rumours maki_thompson: The Maki-Thompson rumour system started by one spreader, which leaves about 20% of people ignorant.
• Manna sandpile manna_sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• Nagel-Schreckenberg nagel_schreckenberg: The Nagel-Schreckenberg traffic system on a ring road, where jams emerge above a critical density.
• Noisy automata noisy_automata: An elementary cellular automaton whose outputs are flipped at random with a small probability.
• Probabilistic Boolean network probabilistic_boolean_network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Ring-road driver agents ring_road_drivers: Drivers on a circular highway, where small random errors of speed set off traffic jams.
• Rock-paper-scissors rock_paper_scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.
• Self-organized branching process self_organized_branching: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
• Stochastic Schelling stochastic_schelling: The stochastic Schelling system of agents moving between blocks, which segregates above beta = 2.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.

stochastic climate

• Hasselmann stochastic climate hasselmann: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.

stochastic escape

• Adaptive climbing in finite populations adaptive_climbing: A finite population that climbs a random fitness landscape by mutation and strong selection.

stochastic map

• Activity-driven temporal networks activity_driven_epidemics: SIS spreading on an activity-driven temporal network, whose edges last one step.
• Huepe-Aldana noisy majority Boolean network noisy_boolean_network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Jain-Krishna autocatalytic networks jain_krishna: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.
• Record-dynamics epidemic of resistant residents record_dynamics_epidemic: Yearly strains that infect every resident less resistant than their strength, who become immune.
• Sequentially linear ecosystem sequentially_linear_ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.
• Solé-Manrubia extinctions sole_manrubia: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Superdiffusive Levy flight levy_flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

stochastic simulation

• Stochastic and continuous Petri nets petri_net: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.

stock market

• Fundamentalists and chartists in an order book order_book: A stock market of fundamentalists and chartists on a grid who trade in a limit order book and imitate.
• Order book with a guru communication network guru_order_book: The order-driven market of Grilli and Tedeschi, whose noise traders imitate gurus they choose by wealth.

stock-flow consistent

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.
• Metaphoria mortal hunter-gatherer and monetary economies metaphoria: Gooding's Metaphoria, a mortal population of hunter-gatherers or of a monetary economy.
• Stock-flow consistent monetary economy stock_flow_consistent_economy: A stock-flow consistent economy, a closed circuit of money between households, firms, and a bank.

stocks and flows

• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.
• Bathtub bathtub: A bathtub, one stock of water with an inflow from the faucet and an outflow through the drain.
• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Cascade classifier cascade_classifier: A cascade classifier, a column of stages that separates a narrow size class of particles by an air flow.
• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Cooling coffee cooling_coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).
• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.
• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.
• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.
• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.
• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.
• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.
• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Penaeid shrimp growth penaeid_shrimp: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.
• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.
• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.
• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.
• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs adaptive_response_costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.
• Stock-flow consistent monetary economy stock_flow_consistent_economy: A stock-flow consistent economy, a closed circuit of money between households, firms, and a bank.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.
• Thermostat thermostat: A room heated by a furnace under a thermostat while heat leaks to the outside.

stokes flow

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Squirmers squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.

stop-and-go

• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.

storage

• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.

strange attractor

• Fractional Lorenz fractional_lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
• Hyperchaotic four-dimensional Rössler flow hyperchaotic_rossler: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
• Hénon map henon_map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
• Langford attractor langford: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
• Lorenz lorenz: The Lorenz-63 system of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
• Lozi map lozi_map: The Lozi map x' = 1 + y - a |x|, y' = b x, the piecewise linear Hénon map with a strange attractor.
• Rössler rossler: The Rössler system, a chaotic flow with one folded band for its standard parameters.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.
• Thomas cyclically symmetric attractor thomas: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.

strong selection weak mutation

• Strong-selection weak-mutation walk on fitness landscapes origin_fixation_walk: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.

structures

• Fuzzy oil drop system of protein structure fuzzy_oil_drop: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.

subdiffusion

• Continuous-time random walk continuous_time_random_walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.

subshift

• Context-free subshift context_free_subshift: A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.
• Even and S-gap subshifts gap_subshift: The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.
• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.
• Subshifts subshifts: A Markov chain on the golden mean shift, the configurations of a ring that never hold two adjacent ones.

substitution

• Cantor cantor_set: The Cantor set as the substitution of every interval by its two outer parts.
• Morse subshift morse_subshift: The Thue-Morse substitution 0 → 01, 1 → 10, whose words make the language of the Morse subshift.
• Sierpiński substitutions sierpinski_substitution: The Sierpiński triangle or carpet as the substitution of a polygon by scaled copies of itself.

succession

• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.

sugarscape

• Sugarscape sugarscape: The Sugarscape of Epstein and Axtell, foragers of random vision and metabolism on two sugar peaks.

superdiffusion

• Lévy walk levy_walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).
• Superdiffusive Levy flight levy_flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

superposition

• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.

superstatistics

• Network superposition model of Erdős-Rényi subgraphs network_superposition: The network superposition model, a union of Erdős-Rényi graphs whose linking probabilities are drawn.

supply chain

• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.
• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.

supply chains

• Namrata's Family, diffusion of working conditions in supply chains supply_chain_diffusion: Chesney's Namrata's Family, buyers who push good working conditions down their supply network.

surface tension

• Bénard-Marangoni convection with a free, thermocapillary surface benard_marangoni: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.

surjective

• Group-valued difference automaton group_difference_automaton: The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.

surrogate

• Model predictive control of Duffing, Van der Pol, and a bilinear DC motor through Koopman surrogates koopman_model_predictive_control: A plant that a discrete-time controller steers with model predictive control on a fitted surrogate.

suspension

• Quarter-car suspension quarter_car: A quarter-car suspension driving over a road step.

swarm control

• Strömbom shepherding of a flock shepherding: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.

swarmalators

• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.

swarming

• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.

swarms

• Bird swarm with pair potentials bird_swarm: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Light-emitting robots with sensory delay light_emitting_robots: Light-emitting robots, which cluster when they sense the light of the others late and spread when early.
• Swarmtaxis swarmtaxis: The swarmtaxis algorithm, which moves a swarm of robots towards a beacon that only some of them see.

symmetric group

• Group-valued difference automaton group_difference_automaton: The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.

symmetry breaking

• Binary ant decision binary_ant_decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Customers choosing between two shops of perishable goods perishable_goods_duopoly: Customers who choose between two shops of perishable goods, preferring the one that sold fresher goods.

synapse

• Adaptrode synapse adaptrode: An Adaptrode synapse driven by a train of input pulses that stops after a while.

synchronization

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters adaptive_kuramoto: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Adler equation adler_equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps aggregate_averaging_synchronization: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Coupled Colpitts oscillators coupled_colpitts: Two detuned chaotic Colpitts oscillators coupled through the difference of their second variables.
• Coupled Rössler and the master stability function coupled_rossler: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.
• Coupled map lattices on networks coupled_map_lattices_on_networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Delayed Kuramoto delayed_kuramoto: Kuramoto phase oscillators coupled all to all through signals that arrive after a delay.
• Discrete-time SIRS with a recovery delay delayed_sirs: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant topological_kuramoto: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Forced Van der Pol forced_van_der_pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Globally coupled Stuart-Landau oscillators with frequency spread oscillator_death: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales heterogeneous_global_coupling: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Huygens clocks and pendulums on coupled supports huygens_clocks: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Kuramoto kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Kuramoto on networks kuramoto_on_networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Kuramoto oscillators with three-body coupling hypergraph_kuramoto: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Kuramoto-Sakaguchi chimera kuramoto_sakaguchi_chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• LEGION oscillators legion: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Mirollo-Strogatz pulse-coupled oscillators mirollo_strogatz: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.
• Nonlocal ring chimera nonlocal_ring_chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.
• Ott-Antonsen reduction ott_antonsen: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.
• Substance exchange between logistic cells in a closed contour closed_contour_cell_exchange: The exchange of signalling molecules between logistic cells arranged in a closed contour.
• Swarmalators swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.
• Swing-equation power grid with Braess paradox power_grid: A ring power grid in the swing-equation system, where more transmission capacity can break synchrony.
• Winfree system winfree: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

synchronous

• Hegselmann-Krause hegselmann_krause: The Hegselmann-Krause system, whose opinions freeze within a few steps into clusters about 2 ε apart.

synthetic biology

• Genetic toggle switch toggle_switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Repressilator repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.

system a

• Allen-Cahn non-conserved relaxation (System A) allen_cahn: The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.

system b

• Active System B+ of motility-induced phase separation active_system_b: Active System B+, the conserved field theory of the density of active particles.
• Cahn-Hilliard phase separation (System B) cahn_hilliard: The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.

system dynamics

• Bass diffusion bass_diffusion: The Bass diffusion system, people who adopt a new product through advertising and word of mouth.
• Bathtub bathtub: A bathtub, one stock of water with an inflow from the faucet and an outflow through the drain.
• Beer distribution game beer_game: Sterman's beer distribution game, a supply chain whose orders swing wider at every echelon upstream.
• Capital stock capital_stock: Meadows' capital stock, grown by reinvested output and drained by depreciation.
• Car-dealer inventory with delays car_dealer: Meadows' car dealership, whose inventory oscillates after a step in demand because of its delays.
• Cocoa cocoa: A cocoa production system, plantations that boom with their profit and bust with the insects they invite.
• Commodity cycle with capacity delay commodity_cycle: A commodity market whose price and capacity cycle because new capacity arrives after a delay.
• Crop livestock household crop_livestock_household: Bala's crop livestock model, a farming household that adds livestock rearing to its crop income.
• Declining energy return on investment energy_return: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.
• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.
• Food relief system food_relief: A food relief system, a pipeline that ships food to a warehouse and distributes it to people in need.
• Food security systems food_security: A food security system, the rice self-sufficiency of a country under subsidy, research, and extension.
• Grain storage system grain_storage: A retail grain store, whose orders through a supply line make its stock oscillate.
• Hilsa fishery hilsa_fishery: A hilsa fishery, the life cycle of a migratory fish that the catch of juveniles and spawners depletes.
• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.
• Mangrove succession mangrove_succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Oil economy oil_economy: Meadows' oil economy, capital that grows on a nonrenewable resource and collapses as it runs out.
• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.
• Population and pollution population_pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Renewable fishery with capital fishery_capital: Meadows' fishing fleet, capital that grows on a renewable resource and can overshoot it.
• Rice land, population, and food security rice_food_balance: Bala's food security model, rice land that grows towards the land available against a growing population.
• Rice milling system rice_milling: A rice milling system, a mill that buys paddy, mills it into rice and by-products, and sells them.
• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.
• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.

system traps

• Drift to low performance drift_to_low_performance: Meadows' drift to low performance trap, eroding goals that follow a pessimistic view of the performance.
• Escalation escalation: Meadows' escalation trap, two parties that each want to exceed the other's stock by a margin.
• Policy resistance policy_resistance: Meadows' policy resistance trap, one stock pulled by three actors toward inconsistent goals.
• Shifting the burden shifting_the_burden: Meadows' shifting the burden trap, a quick fix that relieves a symptom and erodes the system's capability.
• Success to the successful success_to_the_successful: Meadows' success to the successful trap, two competitors that share a budget by their past success.

systemic risk

• DebtRank cascades debtrank: The DebtRank cascade of distress from a set of banks through their interbank loans.
• Default cascades with thresholds and recovery default_cascade: A cascade of defaults through interbank exposures, from a shock to the external assets of some banks.
• Eisenberg-Noe clearing of interbank payments eisenberg_noe_clearing: The clearing of interbank payments of Eisenberg and Noe, a fixed point of proportional payments.
• Fire sales of overlapping portfolios overlapping_portfolio_fire_sales: Banks with overlapping portfolios whose fire sales of common assets spread their losses to one another.

takeover

• Employment takeover employment_takeover: The employment takeover system on a periodic checkerboard, with half of each kind coloured at random.

targeted therapy

• Paclitaxel and trastuzumab with antibody binding her2_targeted_therapy: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.

taxis

• Swarmtaxis swarmtaxis: The swarmtaxis algorithm, which moves a swarm of robots towards a beacon that only some of them see.

technology

• Arthur-Polak technology evolution arthur_polak: Arthur and Polak's evolution of technology, which builds logic functions from NAND gates.
• NK technology system nk_technology: McNerney's system of technological improvement, whose cost falls as a power of the cumulative production.

temporal network

• Activity-driven temporal networks activity_driven_epidemics: SIS spreading on an activity-driven temporal network, whose edges last one step.
• Echo chambers and radicalization on an activity-driven network echo_chamber_radicalization: The radicalization model of echo chambers of Baumann and coauthors on an activity-driven network.
• Nonlinear three-body consensus on temporal hypergraphs hypergraph_consensus: The nonlinear three-body consensus system on a random hypergraph of triangles that switch on and off.
• SIS on partnerships that dissolve and re-form pair_formation_sis: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.

tensor

• Active nematic and polar gels active_gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Deformable active particles of Ohta and Ohkuma deformable_active_particles: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.
• Dry active nematics dry_active_nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.

thermal

• Heated room with a thermostat heated_room: A room assembled from thermal and electrical components, heated under a thermostat with hysteresis.

thermohaline circulation

• Stommel thermohaline boxes stommel: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.

thermostat

• Nosé-Hoover oscillator nose_hoover: The Nosé-Hoover oscillator x' = y, y' = y z - x, z' = 1 - y², with tori next to a chaotic sea.

theta neurons

• Theta neurons with spike-timing-dependent plasticity stdp_network: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.

theta-gamma

• Next-generation neural mass with theta forcing next_generation_neural_mass: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.

third order

• Sprott jerk sprott_jerk: Sprott's jerk equation x''' = -A x'' - x' + |x| - 1, chaotic near A = 0.6.

thomas

• Generalised multi-valued logical networks generalized_logical_network: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.

three dimensions

• Boids in three dimensions boids_3d: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions anisotropic_brownian_particles: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.

threshold

• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis coagulation_positive_feedback: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.
• Threshold adoption threshold_adoption: The threshold adoption system in social circles, started by the 2.5% of agents with the lowest thresholds.
• Watts threshold cascades watts_cascades: The Watts threshold system of global cascades started by a few random seeds.

tipping

• Rosenzweig-MacArthur and rate tipping rosenzweig_macarthur: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.
• Stommel thermohaline boxes stommel: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.

tipping point

• Fair Food Program fair_food_program: Chesney's Fair Food Program, in which pressure on tomato buyers draws farms into fair wages.
• Intrinsic-instability populations intrinsic_instability: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.

tipping points

• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Leading-following tipping cascade leading_following_cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Network of interacting tipping elements tipping_cascade_network: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping sutera_fraedrich_tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.
• Stochastic fold with a ramped control parameter ramped_stochastic_fold: A stochastic fold whose control is ramped through the tipping point of its lower branch.

tissue

• Cellular Potts model of cell sorting on square, triangular, and disordered lattices cellular_potts: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.
• Szabó cell system szabo_cells: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.

torus

• Langford attractor langford: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
• Sprott flow with a conservative torus and a chaotic attractor sprott_mixed_flow: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.

toxicity

• Cumulative drug toxicity drug_toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• Resistance subpopulations under three drugs multidrug_resistance: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.

trade

• Bilateral two-good trade on complete, isolated, and small-world networks wilhite_trade_networks: Wilhite's bilateral barter of two goods among traders on one of four trade networks.

traffic

• Asymmetric exclusion process with open boundaries asep: The asymmetric simple exclusion process on an open line, in continuous time.
• Car following with delay delayed_car_following: A chain of cars that react to the speed difference to the car ahead after a reaction time.
• Charbonneau traffic charbonneau_traffic: Charbonneau's traffic system, where random braking of single drivers sets off traffic jams.
• GKT non-local traffic gkt_traffic: The gas-kinetic-based traffic system, where drivers brake on the traffic at an interaction point ahead.
• Intelligent driver system intelligent_driver_system: The intelligent driver system of cars on a ring road.
• Kerner-Konhäuser system kerner_konhauser: The Kerner-Konhäuser system, where viscous traffic flow forms dense moving clusters.
• Langevin car-following toward a density-dependent speed langevin_traffic: Cars on a ring road whose speeds relax with noise towards a desired speed set by the local density.
• Lighthill-Whitham-Richards lighthill_whitham_richards: The Lighthill-Whitham-Richards system, the conservation of cars with the Greenshields speed law.
• Moment equations of Langevin traffic with a jam instability traffic_density_equation: The density and flux equations of Langevin traffic on a ring road, whose uniform flow breaks into jams.
• Nagel-Schreckenberg nagel_schreckenberg: The Nagel-Schreckenberg traffic system on a ring road, where jams emerge above a critical density.
• Optimal velocity optimal_velocity: The optimal velocity system of cars on a ring road, where uniform flow breaks into stop-and-go waves.
• Payne system payne: The Payne system, where the speed of traffic relaxes to the Greenshields speed and anticipates density.
• Ring-road driver agents ring_road_drivers: Drivers on a circular highway, where small random errors of speed set off traffic jams.
• TASEP tasep: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.

trafficking

• Brothel movement system trafficking: The brothel movement system, where victims moved between brothels leave traces in bills and train trips.

trail formation

• Active walkers and trail formation active_walkers: Active walkers who walk between the corners of a green and wear a system of trails into it.

transient attractors

• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.

transport

• SIR on a network as an agent system network_sir: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.

trap system

• Bouchaud trap system trap_system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.

traps

• Fully biased walk among quenched broad traps biased_trap_walk: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.

traveling waves

• Min-protein skeleton system as mass-conserving reaction-diffusion min_protein_skeleton: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Nonreciprocal Cahn-Hilliard nonreciprocal_cahn_hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.

tree

• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• Graphlet arrival growth graphlet_arrival: Growth by the arrival of three-node graphlets, a tree whose degree exponent rises with alpha.

tree automorphism

• Adding machine (odometer) adding_machine: The adding machine r → r + 1 on the q-adic integers, the odometer acting on the q-ary tree.

triad formation

• Holme-Kim growth holme_kim: Holme-Kim growth, scale-free like Barabási-Albert growth but with clustering that stays high.

triadic closure

• Davidsen-Ebel-Bornholdt network of triadic closure and node turnover davidsen_ebel_bornholdt: The Davidsen-Ebel-Bornholdt network, whose triadic closure and node turnover give high clustering.

triadic interactions

• Spatial triadic percolation on a torus spatial_triadic_percolation: Spatial triadic percolation, where nearby signed regulators make the giant component form moving patterns.
• Triadic percolation triadic_percolation: Triadic percolation, where regulating nodes switch links on and off from the giant component.

tumour

• Immune competition immune_competition: The kinetic system of the competition between tumour cells and immune cells.

tumour growth

• Exponential-linear tumour growth exponential_linear_growth: A tumour that grows exponentially while small and linearly once its growth reaches a limiting rate.
• Generalized logistic (Richards) growth generalized_logistic: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.
• Gompertz growth gompertz: A tumour whose growth slows as the logarithm of the room left, dn/dt = r n log(K / n).
• Gompertz growth with a vascular carrying capacity dynamic_carrying_capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.
• Power-law tumour growth power_law_growth: A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.
• Von Bertalanffy growth von_bertalanffy: A tumour that grows by the balance of anabolism and catabolism, dn/dt = r n^p - d n.

tumour-immune

• Normal, tumour, and immune cells under chemotherapy de_pillis_radunskaya: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Resting and hunting immune cells with a conversion delay under chemotherapy delayed_resting_hunting_chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.

turbulence

• Decaying turbulence decaying_turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.

turing machine

• Turing machine (three-state busy beaver) turing_machine: A Turing machine, a head with an internal state that reads, writes, and moves on a tape of symbols.

turtle

• Koch koch_curve: The Koch curve as the L-system F → F+F--F+F drawn by a turtle.

two-level replicator

• Two-level replicator equation of additive pairwise group conflict two_level_replicator: The two-level replicator equation of defectors and altruistic punishers, additive pairwise group conflict.

two-phase flow

• Two-phase zone exchange two_phase_zone_exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.

undecided

• Mean-field flows of undecided agents undecided_flows: The mean-field flows between agents against, A, undecided, C, and for, B.
• Three-state opinion with undecided agents three_state_opinion: The three-state opinion system of agents against, undecided, and for, on a random regular network.

unemployment

• Mark-0 economy with a Covid shock and policies mark0: The Mark-0 economy of Sharma, Bouchaud, and co-authors, with a Covid shock to supply and demand.

unstable equilibrium

• PID-stabilised inverted pendulum pid_inverted_pendulum: An inverted pendulum on a cart that a PID controller of its angle keeps upright.

urban

• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.

urban scaling

• Gravity interaction system of urban scaling gravity_urban_scaling: The residents of a city at uniform positions in a disc, each interacting with the others as r^-gamma.
• Urban scaling of interactions urban_scaling_interactions: A city that grows by residents and internal links, so its meetings grow faster than its population.

urn

• Interacting urns with triggering on a network interacting_urns_network: One urn with triggering per node of a network, which draws from its own urn and those of its neighbours.
• Pólya urn polya_urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.

vaccination

• SIR with optimal vaccination sir_vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.

vaccine

• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination dendritic_cell_vaccine: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour-immune populations with division delays under dendritic-cell vaccination delayed_dc_vaccine: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.

vasculature

• Vessel conductances adapting to Kirchhoff flows adaptive_flow_network: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.

vegetation

• Daisyworld daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.
• Forest dieback forest_dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Forest dieback with replicator dynamics of mitigators social_forest_dieback: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.

vehicles

• Regenerative braking regenerative_braking: A car braked by a Faraday disk generator on its wheel, which charges a supercapacitor.
• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.

vibration

• Electromagnetic vibrating table vibrating_table: An electromagnetic vibrating table, a shaker driven by a sinusoidal supply after its switch closes.
• Vibration energy harvester vibration_energy_harvester: A dual-mass piezoelectric harvester in a car suspension, driven by the roughness of the road.
• Wheel and rail contact wheel_rail_contact: A rail segment on a sleeper, loaded by the wheels of passing bogies.

vicsek

• Nonreciprocal Vicsek system nonreciprocal_vicsek: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.

virotherapy

• Glioma, glia, and neurons under temozolomide and virus glioma_chemovirotherapy: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Oncolytic adenovirus with T cells and dendritic-cell injection oncolytic_virus_dendritic: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• Uninfected and infected cells, virus, and chemotherapy chemovirotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.
• Uninfected and infected tumour cells under an oncolytic virus oncolytic_virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.

viscoelastic

• One-dimensional active viscoelastic and poroelastic gel with contractile cells active_viscoelastic_gel: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.

vortices

• XY xy: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.

voter

• Coevolving voter coevolving_voter: The coevolving voter system, which fragments into consensual components when rewiring is fast.

voting

• Majority rule automaton majority_rule: Voters that take the majority opinion of their neighbourhood, which coarsens into domains.
• Voter variants voter_variants: Voter system variants on a network, with threshold voters and a share of contrarians.

wages

• Ajani's Choice ajanis_choice: Chesney's Ajani's Choice, migrant farm workers and employers among whom underpayment emerges.

waiting times

• Poisson process poisson_process: Independent Poisson processes, counters that grow by one at exponentially distributed waiting times.

waning immunity

• SIRS sirs: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.

waste management

• Solid waste system solid_waste: A solid waste system, the waste of a growing city collected, recycled, composted, and landfilled.

water resources

• Irrigation system irrigation: An irrigation system, a reservoir and an aquifer that keep the root zone of the crops supplied.

wave packet

• Schrödinger equation minimally coupled to an electromagnetic potential charged_schrodinger: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.

waves

• Schrödinger equation as real and imaginary fields schrodinger: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Shallow-water dam break shallow_water: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Spring chains and guitar string spring_chain: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Standing waves on a string standing_waves: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.
• Wave-number conservation and wave breaking wave_breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

wealth

• Boltzmann wealth exchange boltzmann_wealth: A Boltzmann wealth exchange system where every agent starts with one unit.
• Bouchaud-Mézard wealth bouchaud_mezard_wealth: Bouchaud and Mezard's wealth dynamics, random multiplicative returns with exchange between neighbours.
• Exchange with debt limit and with saving propensity kinetic_wealth_exchange: A kinetic wealth exchange system with a saving propensity or a debt limit.
• Kinetic wealth and welfare dynamics kinetic_wealth_welfare: The kinetic theory of wealth classes, with a welfare policy when the tax is positive.
• Sugarscape sugarscape: The Sugarscape of Epstein and Axtell, foragers of random vision and metabolism on two sugar peaks.

welfare

• Kinetic wealth and welfare dynamics kinetic_wealth_welfare: The kinetic theory of wealth classes, with a welfare policy when the tax is positive.

western boundary current

• Quasi-geostrophic double gyre double_gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.

winner-take-all

• Fukai-Tanaka winner-take-all network winners_take_all: A winner-take-all network of units with self-limited growth and mutual inhibition.

winners-take-all

• Dense homogeneous associative network of transient attractors associative_thought_process: The dense homogeneous associative network of Gros, whose memories become active one after the other.

working memory

• Amari neural field amari_neural_field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.

yang-mills

• Lattice Yang-Mills and pure lattice QCD lattice_yang_mills: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.

zealots

• Majority rule with willingness and zealots willing_majority: The majority rule with willingness, where zealots of willingness 0 never change their opinion.

zero intelligence

• Zero-intelligence double auction double_auction: Gode and Sunder's double auction of zero-intelligence traders, a step per trading period.

zero-range process

• Dissipative zero-range process on a tree dissipative_zero_range: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• Zero-range process zero_range_process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

zipf

• Miller's random typewriting random_typewriting: Miller's random typewriting, monkeys whose random keystrokes give Zipf's law of word frequencies.
• Sample-space-reducing process sample_space_reducing: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.

zombies

• Zombie outbreak of susceptible, zombie, and removed compartments zombie_outbreak: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.