Rucklidge

Systems / ODEs / Chaotic attractors

A model of convection in an imposed magnetic field — a compact chaotic flow.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + b y - y z \\ \dot{y} &= x \\ \dot{z} &= y^{2} - z \end{aligned} \]

Parameters

Symbol	Default	Role
a	2	dissipation parameter
b	6.7	forcing (Rayleigh-like) 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

$\nabla\!\cdot f = - a - 1$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Rucklidge()
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

Rucklidge (1992), J. Fluid Mech. 237, 209-229
doi:10.1017/s0022112092003392

BibTeX

@misc{rucklidge,
  title = {Rucklidge system},
  note = {Rucklidge (1992), J. Fluid Mech. 237, 209-229},
  doi = {10.1017/s0022112092003392}
}