HyperBao

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow formed by adding a linear feedback controller to the Lü system, giving two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	36	coupling rate of the x–y (Lü) subsystem
b	3	damping of the z mode
c	20	self-gain of the y mode
d	0.1	gain of the y·z term feeding the auxiliary state w
e	21	feedback gain of x onto w

State variables: x, y, z, w

Properties

Lyapunov spectrum

≥ 2 positive exponents (literature — hyperchaotic: two positive exponents)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperBao()
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

Bao & Liu (2008), Chin. Phys. Lett. 25, 2396-2399
doi:10.1088/0256-307x/25/7/018

BibTeX

@misc{hyperbao,
  title = {HyperBao system},
  note = {Bao & Liu (2008), Chin. Phys. Lett. 25, 2396-2399},
  doi = {10.1088/0256-307x/25/7/018}
}