Chirikov¶

Systems / Maps / Chaotic maps

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

discrete · map2 dimensionsmixedconservative

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

State variables: p, x

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

276 fixed points
0 stable · 276 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Chirikov()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Chirikov (1979), Phys. Rep. 52, 263-379
doi:10.1016/0370-1573(79)90023-1

BibTeX

@misc{chirikov,
  title = {Chirikov system},
  note = {Chirikov (1979), Phys. Rep. 52, 263-379},
  doi = {10.1016/0370-1573(79)90023-1}
}