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Mathematics

The registered systems of skcomplex.systems.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 system 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.
  • Cascais-Dilão-Noronha Gaussian map: The Cascais-Dilão-Noronha Gaussian map x' = exp(-alpha² x²) + β of an interval.
  • 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.
  • Conservative Hénon map: Hénon's area-preserving quadratic map, a rotation by 2π nu with a quadratic kick.
  • 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.
  • 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: 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.
  • Hyperchaotic four-dimensional Rössler flow: Rössler's four-dimensional flow, hyperchaotic with two positive Lyapunov exponents for its defaults.
  • Hénon map: The Hénon map x' = 1 - a x² + y, y' = b x, with a fractal strange attractor for its defaults.
  • Ikeda delay equation: The Ikeda delay equation x' = a sin x(t - tau) in the form of Dilão (2023, Exercise 6.4).
  • Intermittency maps: The Pomeau-Manneville map x' = x + x^z (mod 1), whose long laminar phases near 0 end in chaotic bursts.
  • Langford attractor: The Langford flow, a rotation whose torus breaks into a strange attractor for the defaults of Dilão.
  • 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 system of convection, chaotic on a butterfly-shaped attractor for its standard parameters.
  • 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's method for z³ = 1 in the complex plane, z' = (2z³ + 1) / (3z²), in real coordinates.
  • Nosé-Hoover oscillator: 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: The quadratic flow x1' = mu x1, x2' = lam (x2 - x1^2), with a slow manifold and a linear embedding.
  • Rössler: The Rössler system, 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 flow with a conservative torus and a chaotic attractor: Sprott's flow with a conservative torus and a dissipative chaotic attractor side by side.
  • 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.
  • Thomas cyclically symmetric attractor: The Thomas flow x' = -a x + sin y with its cyclic permutations, chaotic for a = 0.18.
  • 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).
  • Random complex square-root function system: The random function system z -> ±√(z - omega), whose points settle on the Julia set of z² + omega.
  • 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