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

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} \\ \dot{y_{1}} &= c y_{1} - y_{0} y_{2} + y_{0} \left(- a + c\right) \\ \dot{y_{2}} &= - b y_{2} + y_{0} y_{1} \end{aligned} \]

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

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

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Chen & Ueta (1999), Int. J. Bifurcation Chaos 9, 1465-1466

BibTeX

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