HyperXu

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow from state-feedback control of a chaotic core, with two positive Lyapunov exponents and an analog-circuit realization.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

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	2	gain of the y feedback in the w equation
e	16	strength of the x·z nonlinearity driving y

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.HyperXu()
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

Letellier & Rössler (2007), Scholarpedia 2(8), 1936
doi:10.4249/scholarpedia.1936

BibTeX

@misc{hyperxu,
  title = {HyperXu system},
  note = {Letellier & Rössler (2007), Scholarpedia 2(8), 1936},
  doi = {10.4249/scholarpedia.1936}
}