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¶

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

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

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a + c + d z + e$
state-dependent

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.ZhouChen()
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¶

Zhou & Chen (2004), Int. J. Bifurcation Chaos
doi:10.1142/s0218127404010175

BibTeX

@misc{zhouchen,
  title = {ZhouChen system},
  note = {Zhou & Chen (2004), Int. J. Bifurcation Chaos},
  doi = {10.1142/s0218127404010175}
}