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

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

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

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

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Bao & Liu (2008), Chin. Phys. Lett. 25, 2396-2399

BibTeX

@misc{hyperbao,
  title = {HyperBao system},
  note = {Bao & Liu (2008), Chin. Phys. Lett. 25, 2396-2399}
}