LuChen

Systems / ODEs / Coupled systems

The Lü attractor — a quadratic flow sitting at the transition between the Lorenz and Chen systems yet belonging to neither.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	36	Prandtl-like x–y coupling gain
b	3	linear decay rate of the z channel
c	18	self-feedback gain of the y 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

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

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.LuChen()
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 (2002), Int. J. Bifurcation Chaos 12, 659-661
doi:10.1142/s0218127402004620

BibTeX

@misc{luchen,
  title = {LuChen system},
  note = {Lü & Chen (2002), Int. J. Bifurcation Chaos 12, 659-661},
  doi = {10.1142/s0218127402004620}
}