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¶

\[ \begin{aligned} \dot{x} &= - a x + 10 y z + y \\ \dot{y} &= 5 x z - x - 0.4 y \\ \dot{z} &= b z - 5 x y \end{aligned} \]

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)

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.NewtonLiepnik()
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¶

Leipnik & Newton (1981), Phys. Lett. A 86, 63
doi:10.1016/0375-9601(81)90165-1

BibTeX

@misc{newtonliepnik,
  title = {NewtonLiepnik system},
  note = {Leipnik & Newton (1981), Phys. Lett. A 86, 63},
  doi = {10.1016/0375-9601(81)90165-1}
}