Torus

Systems / ODEs / Oscillatory systems

Quasiperiodic motion on a 2-torus — two incommensurate frequencies, no chaos.

continuous · ODE3 dimensionsquasiperiodic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	0.5	minor radius (tube radius)
n	15.3	tube windings per revolution (irrational = quasiperiodic)
r	1	major radius (centre to tube centre)

Properties

Lyapunov spectrum

$0,\; 0,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Strogatz (1994), Nonlinear Dynamics and Chaos

BibTeX

@misc{torus,
  title = {Torus system},
  note = {Strogatz (1994), Nonlinear Dynamics and Chaos}
}