Chen

Systems / ODEs / Coupled systems

A dual to Lorenz with the opposite sign convention — a generalized chaotic flow in the Lorenz family.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	35	coupling gain of the x-y channel
b	3	linear decay rate of the z channel
c	28	self-feedback gain of the y channel

State variables: x, y, z

Properties

Lyapunov spectrum

$+2.03,\; 0,\; -12.03$
Lü, Chen, Cheng & Čelikovský (2002), Int. J. Bifurcation Chaos 12, 2917-2926

Kaplan–Yorke dimension

$D_{KY} = 2.169$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Chen()
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

Chen & Ueta (1999), Int. J. Bifurcation Chaos 9, 1465-1466
doi:10.1142/s0218127499001024

BibTeX

@misc{chen,
  title = {Chen system},
  note = {Chen & Ueta (1999), Int. J. Bifurcation Chaos 9, 1465-1466},
  doi = {10.1142/s0218127499001024}
}