Zaslavskii¶

Systems / Maps / Chaotic maps

The Zaslavsky dissipative standard map — a damped kicked rotor whose contraction folds the standard-map web onto a fractal strange attractor.

discrete · map2 dimensionschaotic

Definition¶

@staticmethod
def _step(X, eps, nu, r):
    x, y = X
    mu = (1 - np.exp(-r)) / r
    xp = x + nu * (1 + mu * y) + eps * nu * mu * np.cos(2 * np.pi * x)
    # Phase variable lives on the unit circle: wrap mod 1 (floor-mod already
    # yields a result in [0, 1)).
    xp = xp % 1
    yp = np.exp(-r) * (y + eps * np.cos(2 * np.pi * x))
    return xp, yp

Parameters¶

Symbol	Default	Role
`eps`	`9`	kick (perturbation) strength
`nu`	`0.3`	coupling between action and phase advance
`r`	`2`	damping rate (exp(-r) is the per-step action contraction)

State variables: x, y

Properties¶

Lyapunov spectrum

$+1.342,\; -3.342$
measured on the strange attractor; the sum is exactly -r = -2 (det J = exp(-r))

Kaplan–Yorke dimension

$D_{KY} = 1.402$

Divergence ∇·f

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

Equilibria

6 fixed points
0 stable · 6 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Zaslavsky (1978), Phys. Lett. A 69, 145-147
doi:10.1016/0375-9601(78)90195-0

BibTeX

@misc{zaslavskii,
  title = {Zaslavskii system},
  note = {Zaslavsky (1978), Phys. Lett. A 69, 145-147},
  doi = {10.1016/0375-9601(78)90195-0}
}