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`	`5`	kick (perturbation) strength
`nu`	`0.2`	coupling between action and phase advance
`r`	`2`	damping rate (exp(-r) is the per-step action contraction)

Properties¶

Lyapunov spectrum

$-0.9976,\; -1.002$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 0$

Divergence ∇·f

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

Equilibria

2 fixed points
0 stable · 2 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Zaslavsky (1978), Phys. Lett. A 69, 145-147

BibTeX

@misc{zaslavskii,
  title = {Zaslavskii system},
  note = {Zaslavsky (1978), Phys. Lett. A 69, 145-147}
}