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

\[ \begin{aligned} \dot{y_{0}} &= g y_{0} + y_{1} \left(y_{0}^{2} + y_{2} - 1\right) \\ \dot{y_{1}} &= g y_{1} + y_{0} \left(- y_{0}^{2} + 3 y_{2} + 1\right) \\ \dot{y_{2}} &= - 2 y_{2} \left(a + y_{0} y_{1}\right) \end{aligned} \]

Parameters

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

Properties

Lyapunov spectrum

$+0.1925,\; +0.001823,\; -0.6544$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.297$

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

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}
}