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

\[ \begin{aligned} \dot{x} &= - \mu x + y z \\ \dot{y} &= - a x - \mu y + x z \\ \dot{z} &= - x y + 1 \end{aligned} \]

Parameters

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

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.RikitakeDynamo()
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

Rikitake (1958), Proc. Cambridge Philos. Soc. 54, 89-105
doi:10.1017/s0305004100033223

BibTeX

@misc{rikitakedynamo,
  title = {RikitakeDynamo system},
  note = {Rikitake (1958), Proc. Cambridge Philos. Soc. 54, 89-105},
  doi = {10.1017/s0305004100033223}
}