HyperWang

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow with a Lorenz-like core, a quadratic x² drive on the z mode, and a fourth feedback state — two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + a y \\ \dot{y} &= b x + w - x z \\ \dot{z} &= - c z + e x^{2} \\ \dot{w} &= - d x \end{aligned} \]

Parameters

Symbol	Default	Role
a	10	coupling rate of the x–y subsystem
b	40	linear gain in the y equation
c	2.5	damping of the z mode
d	10.6	feedback gain of x onto the auxiliary state w
e	4	strength of the x² nonlinearity driving z

State variables: x, y, z, w

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a - c$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperWang()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

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

Reference

Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39
doi:10.1590/s0103-97332009000500007

BibTeX

@misc{hyperwang,
  title = {HyperWang system},
  note = {Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39},
  doi = {10.1590/s0103-97332009000500007}
}