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¶

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} + h y_{1} + s y_{1} y_{2} \\ \dot{y_{1}} &= - b y_{1} - p y_{0} + q y_{0} y_{2} \\ \dot{y_{2}} &= c y_{2} - r y_{0} y_{1} \end{aligned} \]

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

Properties¶

Lyapunov spectrum

$+0.1607,\; -0.07699,\; -16.68$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.005$

Divergence ∇·f

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

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Liu & Chen (2004), Int. J. Bifurc. Chaos 14, 1395-1403

BibTeX

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