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¶

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} - y_{1} y_{2} \\ \dot{y_{1}} &= b y_{1} + y_{0} y_{2} \\ \dot{y_{2}} &= c y_{2} + \frac{y_{0} y_{1}}{3} \end{aligned} \]

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

Properties¶

Lyapunov spectrum

$+0.1943,\; -0.01383,\; -5.56$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.032$

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Chen & Lee (2004), Chaos Solitons Fractals 21, 957-965

BibTeX

@misc{chenlee,
  title = {ChenLee system},
  note = {Chen & Lee (2004), Chaos Solitons Fractals 21, 957-965}
}