Thomas

Systems / ODEs / Chaotic attractors

Cyclically symmetric flow whose 27 equilibria and labyrinthine trajectories arise from pure sinusoidal feedback.

continuous · ODE3 dimensionschaoticcyclically symmetric

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Definition

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + b \sin{\left(y_{1} \right)} \\ \dot{y_{1}} &= - a y_{1} + b \sin{\left(y_{2} \right)} \\ \dot{y_{2}} &= - a y_{2} + b \sin{\left(y_{0} \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1.85	dissipation (friction) coefficient (smaller = more chaotic)
b	10	amplitude of the sinusoidal forcing

Properties

Lyapunov spectrum

$+0.633,\; -0.01291,\; -6.17$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.1$

Divergence ∇·f

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

Equilibria

18 equilibria
0 stable · 18 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Thomas()
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{thomas,
  title = {Thomas system},
  note = {Thomas (1999), Int. J. Bifurc. Chaos 9, 1889-1905}
}