Circle map

Summary

The circle map, or standard circle map, is the two-parameter family $\theta \mapsto \theta + \Omega - \frac{K}{2\pi} \sin(2\pi\theta)$ of maps of the circle $\R/\Z$. It models a rotation by $\Omega$ disturbed by a nonlinear kick of strength $K$. It is the basic model of mode locking: its rotation number locks to rational values on the Arnold tongues. For $0 \le K < 1$ it is a family of circle diffeomorphisms, and at $K = 1$ it has a critical point.