Circle¶

Systems / Maps / Geometric maps

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

discrete · map1 dimensionmode-locking

Circle return map · bifurcation diagram — 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)

State variables: theta

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

4 fixed points
0 stable · 4 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Circle()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

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}
}