QiChen

Systems / ODEs / Coupled systems

A Lorenz/Chen-template flow whose extra y*z term gives multiple equilibria and a four-wing chaotic attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	38	x–y channel coupling gain
b	2.666	linear decay rate of the z channel
c	80	forcing gain of x into y

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.QiChen()
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

Qi et al. (2008), Chaos Solitons Fractals 38, 705-721
doi:10.1016/j.chaos.2006.09.012

BibTeX

@misc{qichen,
  title = {QiChen system},
  note = {Qi et al. (2008), Chaos Solitons Fractals 38, 705-721},
  doi = {10.1016/j.chaos.2006.09.012}
}