English: This image shows the winding number for the en:circle map. The circle map is defined by the iteration of the map

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

The winding number is given by considering the asymptotic behavior of the map, that is, by

${\displaystyle \omega =\underset{n\to \mathrm{\infty }}{lim}\frac{{\theta }_n}{n}.}$

In this image, black denotes a winding number of 0, red denotes a winding number of 1, and green denotes a winding number of 0.5. The plot is such that the frequency Ω runs from 0 to 1 along the x-axis, and the coupling constant K runs from 0 at the bottom of the image to 2π at the top.