SprottTorus

Systems / ODEs / Chaotic attractors

A time-reversible equilibrium-free flow that is conservative on nested invariant tori for some initial conditions and hosts a hidden strange attractor with a twin repellor for others.

continuous · ODE3 dimensionsconservative

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= 2 x y + x z + y \\ \dot{y} &= - 2 x^{2} + y z + 1 \\ \dot{z} &= - x^{2} + x - y^{2} \end{aligned} \]

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 2 y + 2 z$
state-dependent

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.SprottTorus()
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

Sprott (2014), Phys. Lett. A 378, 1361-1363
doi:10.1016/j.physleta.2013.11.004

BibTeX

@misc{sprotttorus,
  title = {SprottTorus system},
  note = {Sprott (2014), Phys. Lett. A 378, 1361-1363},
  doi = {10.1016/j.physleta.2013.11.004}
}