LiuChen¶

Systems / ODEs / Coupled systems

A generalized-Lorenz double-scroll flow (Sakarya family) at the Liu–Chen parameters, keeping two quadratic cross-products with no linear cross-coupling.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= a x + h y + s y z \\ \dot{y} &= - b y - p x + q x z \\ \dot{z} &= c z - r x y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`0.4`	x self-feedback
`b`	`12`	y self-feedback
`c`	`-5`	z self-feedback
`h`	`0`	y→x linear coupling
`p`	`0`	x→y linear coupling
`q`	`-1`	x·z strength
`r`	`1`	x·y strength
`s`	`1`	y·z strength

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.LiuChen()
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¶

Liu & Chen (2004), Int. J. Bifurc. Chaos 14, 1395-1403
doi:10.1142/s0218127404009880

BibTeX

@misc{liuchen,
  title = {LiuChen system},
  note = {Liu & Chen (2004), Int. J. Bifurc. Chaos 14, 1395-1403},
  doi = {10.1142/s0218127404009880}
}