ZhouChen¶

Systems / ODEs / Coupled systems

A Zhou–Chen quadratic flow carrying paired y*z nonlinearities alongside x*z and x*y terms, producing a chaotic attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`2.97`	self-feedback gain on the x channel
`b`	`0.15`	linear y-into-x coupling gain
`c`	`-3`	self-feedback gain on the y channel
`d`	`1`	y*z cross-product strength in the y equation
`e`	`-8.78`	self-feedback gain on the z channel

Properties¶

Lyapunov spectrum

$+0.7143,\; -0.0289,\; -11.98$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.057$

Divergence ∇·f

$\nabla\!\cdot f = a + c + d y_{2} + e$
state-dependent

Equilibria

5 equilibria
2 stable · 3 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Zhou & Chen (2004), Int. J. Bifurcation Chaos

BibTeX

@misc{zhouchen,
  title = {ZhouChen system},
  note = {Zhou & Chen (2004), Int. J. Bifurcation Chaos}
}