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

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

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

Properties

Lyapunov spectrum

$+0.9096,\; +0.108,\; +0.01091,\; -13.53$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.076$

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.HyperXu()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Letellier & Rössler (2007), Scholarpedia 2(8), 1936

BibTeX

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