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

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

Properties

Lyapunov spectrum

$+0.007467,\; +0.005587,\; -0.01177$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

$\nabla\!\cdot f = 2 y_{1} + 2 y_{2}$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Sprott (2014), Phys. Lett. A 378, 1361-1363

BibTeX

@misc{sprotttorus,
  title = {SprottTorus system},
  note = {Sprott (2014), Phys. Lett. A 378, 1361-1363}
}