HyperJha

Systems / ODEs / Exotic systems

A 4-D hyperchaotic extension of the Lorenz system with a fourth state coupled back through an x·z term, giving two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	10	Lorenz coupling rate of the x–y subsystem
b	28	Lorenz-like gain in the y equation
c	2.667	damping of the z mode
d	1.3	self-gain of the auxiliary feedback state w

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

$\nabla\!\cdot f = - a - c + d - 1$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperJha()
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

Meier (2003), Presentation of Attractors with Cinema

BibTeX

@misc{hyperjha,
  title = {HyperJha system},
  note = {Meier (2003), Presentation of Attractors with Cinema}
}