HyperPang

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow built by adding a linear controller to the Lü system, with two positive Lyapunov exponents and a studied Hopf bifurcation.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

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

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	2	feedback gain of x and y onto the auxiliary 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

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

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperPang()
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

Pang & Liu (2011), J. Comput. Appl. Math. 235, 2775
doi:10.1016/j.cam.2010.11.029

BibTeX

@misc{hyperpang,
  title = {HyperPang system},
  note = {Pang & Liu (2011), J. Comput. Appl. Math. 235, 2775},
  doi = {10.1016/j.cam.2010.11.029}
}