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

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

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

Properties

Lyapunov spectrum

$+0.3155,\; +0.2354,\; +0.009597,\; -12.93$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.043$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Meier (2003), Presentation of Attractors with Cinema

BibTeX

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