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Mathematics

The registered models of skcomplex.models.mathematics. The page of each shows its factory, its component types with their states, parameters, and laws as written, and, for the forms that have them, its equations as SymbolicAnalysis returns them, so it serves as the pattern for a model of your own.

Dynamical systems

  • Baker: The generalized baker map, which stretches the unit square along x and stacks the two pieces in y.
  • Bernoulli doubling map: An ensemble of the doubling map x' = 2x (mod 1), iterated exactly on fractions with an odd denominator.
  • Cat map: Arnold's cat map (x, y) → (2x + y, x + y) (mod 1), an area-preserving hyperbolic map of the torus.
  • Circle: The sine circle map θ' = θ + Ω - K / (2π) sin 2πθ (mod 1), which locks into Arnold tongues.
  • Cubic chaotic map and its chaotic walk: Bertin's cubic chaotic map and the chaotic walk it drives, which spreads like a random walk.
  • Cusp normal forms: The cusp normal form x' = β1 + β2 x - x³, bistable inside the cusp 27 β1² < 4 β2³.
  • 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: An ensemble of the skew tent map, whose density relaxes to the uniform invariant density for any a.
  • Hopf: The Hopf normal form, whose focus gives birth to a limit cycle of radius √(-μ / a) as μ crosses 0.
  • Hénon map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
  • Intermittency maps: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
  • Logistic map: The logistic map x_{t+1} = r x_t (1 - x_t), which reaches chaos through period doubling as r grows.
  • Lorenz: The Lorenz-63 model of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
  • Newton-Raphson map: Newton's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.
  • Quadratic system with a slow manifold and a finite Koopman embedding: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.
  • Rössler: The Rössler model, a chaotic flow with one folded band for its standard parameters.
  • Rössler with an Ornstein-Uhlenbeck control parameter: The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
  • Saddle-node: The saddle-node normal form x' = μ - x², whose two equilibria ±√μ meet and vanish at μ = 0.
  • Sprott jerk: Sprott's jerk equation x''' = -A x'' - x' + |x| - 1, chaotic near A = 0.6.
  • Standard: The Chirikov standard map p' = p + K sin θ, θ' = θ + p', with global chaos above K ≈ 0.9716.
  • Subcritical Hopf normal form: The subcritical Hopf normal form, where a stable focus coexists with a stable limit cycle.
  • Takens-Bogdanov: The Takens-Bogdanov normal form x' = y, y' = β1 + β2 x + x² + s x y, where a saddle-node meets a Hopf.
  • Tent: The tent map x' = μ min(x, 1 - x), chaotic with Lyapunov exponent ln μ for μ above 1.
  • Towel maps: Rössler's towel map, a three-dimensional map with two positive Lyapunov exponents.

Fractals

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

Fractional dynamics

  • Fractional Lorenz: The fractional Lorenz flow of Grigorenko and Grigorenko, chaotic for orders close enough to 1.
  • 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: The fractional damped oscillator D^alpha x = v, D^alpha v = -omega^2 x - 2 gamma v.
  • Fractional relaxation: Fractional relaxation D^alpha x = -lambda x, which decays as the Mittag-Leffler function E_alpha.

Networks

Stochastic processes

  • Ageing random walk: Ageing random walks, whose up steps become rarer the more of them a walker has taken.
  • Bernoulli process: Independent Bernoulli processes, one success or failure per step, whose success counts are binomial.
  • Birth-death process and M/M/1 queue: Independent M/M/1 queues, birth-death processes with constant arrival and service rates.
  • Fragmentation: The random fragmentation of an object of integer size, whose fragments keep breaking into smaller ones.
  • Galton-Watson branching: A Galton-Watson branching process with Poisson offspring, one generation per step.
  • Hidden Markov source: A hidden Markov source, whose symbols are emitted at random from the states of a hidden Markov chain.
  • 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: Brownian particles held by a restoring force that weakens with distance.
  • Persistent random walk: Walkers on the integers that keep the direction of their last step with a probability.
  • Poisson process: Independent Poisson processes, counters that grow by one at exponentially distributed waiting times.
  • Pólya urn: Independent Pólya urns, whose share of red balls converges to a random limit with a beta distribution.
  • Random walk with autoregressive velocity: Random walks whose velocity is an autoregressive process, x ← x + v and v ← alpha v + xi.
  • Random walks: Independent walkers on the integers that step at random times, a continuous-time random walk.
  • Sample-space-reducing process: Sample-space-reducing processes, whose state visits follow Zipf's law 1 / i for lam = 1.
  • Stochastic on-off intermittency: Stochastic on-off intermittency, dx/dt = (epsilon + xi) x - x^3 with Stratonovich white noise xi.