NewtonLiepnik

Systems / ODEs / Exotic systems

Euler's rigid-body equations with a linear feedback torque, producing two coexisting strange attractors selected by the initial basin.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + 10 y_{1} y_{2} + y_{1} \\ \dot{y_{1}} &= 5 y_{0} y_{2} - y_{0} - 0.4 y_{1} \\ \dot{y_{2}} &= b y_{2} - 5 y_{0} y_{1} \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.4	linear damping coefficient on the x mode
b	0.175	linear gain on the z mode (the feedback parameter)

Properties

Lyapunov spectrum

$+0.1339,\; +0.005798,\; -0.7647$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.183$

Divergence ∇·f

$\nabla\!\cdot f = - a + b - 0.4$
constant

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Leipnik & Newton (1981), Phys. Lett. A 86, 63

BibTeX

@misc{newtonliepnik,
  title = {NewtonLiepnik system},
  note = {Leipnik & Newton (1981), Phys. Lett. A 86, 63}
}