LuChenCheng¶

Systems / ODEs / Coupled systems

A unified one-parameter family that interpolates continuously between the Lorenz and Chen attractors, staying chaotic across the whole transition.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`-10`	self-feedback gain of the y channel (sets x-channel decay with b)
`b`	`-4`	self-feedback gain of the z channel (sets x-channel decay with a)
`c`	`18.1`	constant forcing on the x channel

Properties¶

Lyapunov spectrum

$+0.2683,\; -0.004583,\; -11.41$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.023$

Divergence ∇·f

$\nabla\!\cdot f = - \frac{a b}{a + b} + a + b$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Lü, Chen & Cheng (2004), Int. J. Bifurcation Chaos 14, 1507-1537

BibTeX

@misc{luchencheng,
  title = {LuChenCheng system},
  note = {Lü, Chen & Cheng (2004), Int. J. Bifurcation Chaos 14, 1507-1537}
}