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¶

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

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

Properties¶

Lyapunov spectrum

$+0.3569,\; -0.01121,\; -21.35$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.016$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Lü & Chen (2002), Int. J. Bifurcation Chaos 12, 659-661

BibTeX

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