File:Devils-staircase.svg

English: This figure shows the winding number for the circle map, as a function of Ω, with K held constant, at K=1. It is an example of the Devil's staircase. Each of the flat regions corresponds to a region of phase-locking; that is, each flat area is a slice through one of the Arnold tongues.

The circle map is a model of the phase-locked loop, and is given by iterating the equation

${\displaystyle \theta _{n+1}=\theta _{n}+\Omega -{\frac {K}{2\pi }}\sin(2\pi \theta _{n}).}$

The winding number is defined as the limiting behavior of the system after many iterations, by:

${\displaystyle \omega =\lim _{n\to \infty }{\frac {\theta _{n}}{n}}.}$

This graph shows ω on the vertical axis as a function of Ω on the horizontal axis.