RabinovichFabrikant

Systems / ODEs / Chaotic attractors

A flow from plasma physics with stiff, spiky excursions and a delicate attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= g x + y \left(x^{2} + z - 1\right) \\ \dot{y} &= g y + x \left(- x^{2} + 3 z + 1\right) \\ \dot{z} &= - 2 z \left(a + x y\right) \end{aligned} \]

Parameters

Symbol	Default	Role
a	1.1	modulation / dissipation balance parameter
g	0.87	linear growth / damping 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

$\nabla\!\cdot f = - 2 a + 2 g$
constant

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.RabinovichFabrikant()
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

Rabinovich & Fabrikant (1979), Sov. Phys. JETP 50, 311-317

BibTeX

@misc{rabinovichfabrikant,
  title = {RabinovichFabrikant system},
  note = {Rabinovich & Fabrikant (1979), Sov. Phys. JETP 50, 311-317}
}