HyperCai

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow augmenting a Lorenz-like chaotic core with a fourth feedback state, yielding two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + a y \\ \dot{y} &= b x + c y + w - x z \\ \dot{z} &= - d z + y^{2} \\ \dot{w} &= - e x \end{aligned} \]

Parameters

Symbol	Default	Role
a	27.5	linear coupling rate of the x–y subsystem
b	3	linear gain in the y equation
c	19.3	linear gain in the y equation
d	2.9	damping of the z mode (fed by y²)
e	3.3	feedback gain of the auxiliary state w onto x

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

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

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperCai()
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

Cai & Huang (2007), Int. J. Nonlinear Sci.

BibTeX

@misc{hypercai,
  title = {HyperCai system},
  note = {Cai & Huang (2007), Int. J. Nonlinear Sci.}
}