Thomas

Systems / ODEs / Chaotic attractors

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

continuous · ODE3 dimensionschaoticcyclically symmetric

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

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

27 equilibria
0 stable · 27 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Thomas()
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

Thomas (1999), Int. J. Bifurc. Chaos 9, 1889-1905
doi:10.1142/s0218127499001383

BibTeX

@misc{thomas,
  title = {Thomas system},
  note = {Thomas (1999), Int. J. Bifurc. Chaos 9, 1889-1905},
  doi = {10.1142/s0218127499001383}
}