RikitakeDynamo

Systems / ODEs / Chaotic attractors

A two-disk dynamo whose chaotic reversals model the geomagnetic field flipping polarity.

continuous · ODE3 dimensionschaoticreversals

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - \mu y_{0} + y_{1} y_{2} \\ \dot{y_{1}} &= - a y_{0} - \mu y_{1} + y_{0} y_{2} \\ \dot{y_{2}} &= - y_{0} y_{1} + 1 \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	asymmetry / coupling between the two disks
mu	1	common damping (dissipation) coefficient

Properties

Lyapunov spectrum

$+0.1189,\; +0.002487,\; -2.121$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.057$

Divergence ∇·f

$\nabla\!\cdot f = - 2 \mu$
constant

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Rikitake (1958), Proc. Cambridge Philos. Soc. 54, 89-105

BibTeX

@misc{rikitakedynamo,
  title = {RikitakeDynamo system},
  note = {Rikitake (1958), Proc. Cambridge Philos. Soc. 54, 89-105}
}