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¶

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

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

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.LuChenCheng()
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¶

Lü, Chen & Cheng (2004), Int. J. Bifurcation Chaos 14, 1507-1537
doi:10.1142/s021812740401014x

BibTeX

@misc{luchencheng,
  title = {LuChenCheng system},
  note = {Lü, Chen & Cheng (2004), Int. J. Bifurcation Chaos 14, 1507-1537},
  doi = {10.1142/s021812740401014x}
}