ChenLee¶

Systems / ODEs / Coupled systems

The governing equations of a gyro with internal-torque feedback, derived from the rigid-body Euler equations, generating a symmetric two-scroll attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= a x - y z \\ \dot{y} &= b y + x z \\ \dot{z} &= c z + \frac{x y}{3} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`5`	self-feedback gain of the x channel
`b`	`-10`	self-feedback gain of the y channel
`c`	`-0.38`	self-feedback gain of the z 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

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.ChenLee()
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¶

Chen & Lee (2004), Chaos Solitons Fractals 21, 957-965
doi:10.1016/j.chaos.2003.12.034

BibTeX

@misc{chenlee,
  title = {ChenLee system},
  note = {Chen & Lee (2004), Chaos Solitons Fractals 21, 957-965},
  doi = {10.1016/j.chaos.2003.12.034}
}