Circle

Systems / Maps / Geometric maps

The circle map — mode-locking, Arnold tongues, and the devil's staircase.

discrete · map1 dimensionmode-locking

return map · bifurcation diagram

Definition

@staticmethod
def _step(X, omega, k):
    theta = X
    thetap = theta + omega + (k / (2 * np.pi)) * np.sin(2 * np.pi * theta)
    return thetap % 1

Parameters

Symbol	Default	Role
omega	0.333	bare winding number (rotation without coupling)
k	5.7	nonlinearity / coupling strength (critical at k=1)

Properties

Lyapunov spectrum

$+1.087$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

4 fixed points
0 stable · 4 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Circle()
traj = sys.iterate(steps=10_000)

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

Reference

Arnold (1965), Amer. Math. Soc. Transl. 46, 213-284

BibTeX

@misc{circle,
  title = {Circle system},
  note = {Arnold (1965), Amer. Math. Soc. Transl. 46, 213-284}
}