Chirikov

Systems / Maps / Chaotic maps

The Chirikov standard map — the paradigm of Hamiltonian chaos, mixing tori and chaotic seas.

discrete · map2 dimensionsmixedconservative

attractor

Definition

@staticmethod
def _step(X, k):
    p, x = X
    pp = p + k * np.sin(x)
    xp = x + pp
    return pp, xp

Parameters

Symbol	Default	Role
k	0.971635	stochasticity (kick) strength

Properties

Lyapunov spectrum

$+0.1336,\; -0.1336$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2$

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

110 fixed points
56 stable · 54 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Chirikov()
traj = sys.iterate(steps=10_000)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Chirikov (1979), Phys. Rep. 52, 263-379

BibTeX

@misc{chirikov,
  title = {Chirikov system},
  note = {Chirikov (1979), Phys. Rep. 52, 263-379}
}