Transferred from en.wikipedia to Commons by Maksim.

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

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

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

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

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.

The en:winding number for the en:circle map. Original artwork by Linas Vepstas en:User:Linas <linas@linas.org>, created 18 January 2006.

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• (del) (cur) 05:42, 19 January 2006 . . en:User:Linas Linas ( en:User_talk:Linas Talk) . . 900x900 (71319 bytes) (The en:Winding_number winding number for the en:Circle_map circle map. Original artwork by Linas Vepstas en:User:Linas User:Linas <linas@linas.org>)