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

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

Parameters

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

Properties

Lyapunov spectrum

$+0.1795,\; -0.004004,\; -3.175$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.055$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Rucklidge (1992), J. Fluid Mech. 237, 209-229

BibTeX

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