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

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} \\ \dot{y_{1}} &= b y_{0} - y_{0} y_{2} + y_{3} \\ \dot{y_{2}} &= - c y_{2} + e y_{0}^{2} \\ \dot{y_{3}} &= - d y_{0} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+1.198,\; +0.04478,\; -0.0003387,\; -13.74$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.09$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.HyperWang()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39

BibTeX

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