ZeraouliaSprott¶

Systems / Maps / Exotic maps

A minimal 2-D rational map whose x-recurrence is a single non-vanishing fraction — one of the simplest rational maps reaching chaos via a quasi-periodic route.

discrete · map2 dimensionschaotic

Definition¶

\[ \begin{aligned} x' &= - \frac{a x}{y^{2} + 1} \\ y' &= b y + x \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`4.8`	gain of the rational x-recurrence (primary control parameter)
`b`	`0.35`	linear feedback coefficient of y

State variables: x, y

Properties¶

Lyapunov spectrum

$+0.189,\; -0.644$
measured (QR tangent-map spectrum, 10**6 iterations, identical from every initial condition tried); the cited a = 2.7 is quasi-periodic

Kaplan–Yorke dimension

$D_{KY} = 1.293$

Divergence ∇·f

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

Equilibria

1 fixed points
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Zeraoulia & Sprott (2011), Int. J. Bifurcation Chaos 21, 155-160
doi:10.1142/s0218127411028325

BibTeX

@misc{zeraouliasprott,
  title = {ZeraouliaSprott system},
  note = {Zeraoulia & Sprott (2011), Int. J. Bifurcation Chaos 21, 155-160},
  doi = {10.1142/s0218127411028325}
}