QiChen¶

Systems / ODEs / Coupled systems

A Lorenz/Chen-template flow whose extra y*z term gives multiple equilibria and a four-wing chaotic attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`38`	x–y channel coupling gain
`b`	`2.666`	linear decay rate of the z channel
`c`	`80`	forcing gain of x into y

Properties¶

Lyapunov spectrum

$+4.038,\; -0.00481,\; -43.7$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.092$

Divergence ∇·f

$\nabla\!\cdot f = - a - b + 1$
constant

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Qi et al. (2008), Chaos Solitons Fractals 38, 705-721

BibTeX

@misc{qichen,
  title = {QiChen system},
  note = {Qi et al. (2008), Chaos Solitons Fractals 38, 705-721}
}