ThomasLabyrinth

Systems / ODEs / Chaotic attractors

Thomas's cyclically symmetric flow at weak damping — a deterministic random walk (labyrinth chaos) across a 3-D lattice of unstable equilibria.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + b \sin{\left(y \right)} \\ \dot{y} &= - a y + b \sin{\left(z \right)} \\ \dot{z} &= - a z + b \sin{\left(x \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.5	dissipation (friction) coefficient
b	10	sinusoidal forcing amplitude

State variables: x, y, z

Properties

Lyapunov spectrum

$+1.229,\; +0.02436,\; -2.753$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.455$

Divergence ∇·f

$\nabla\!\cdot f = - 3 a$
constant

Equilibria

120 equilibria
0 stable · 120 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Thomas (1999), Int. J. Bifurc. Chaos 9, 1889-1905

BibTeX

@misc{thomaslabyrinth,
  title = {ThomasLabyrinth system},
  note = {Thomas (1999), Int. J. Bifurc. Chaos 9, 1889-1905}
}