Image:Circle map winding number.jpeg

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[edit] Description

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

$\theta_{n+1}=\theta_n + \Omega -\frac{K}{2\pi} \sin (2\pi \theta_n).$

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

$\omega=\lim_{n\to\infty} \frac{\theta_n}{n}.$

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.

[edit] Summary

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

[edit] Licensing

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Subject to disclaimers.

This picture/multimedia file is now available on Wikimedia Commons as Circle map winding number.jpeg.
Images which have been tagged with this template may be deleted immediately after satisfying these conditions (CSD I8).

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(del) (cur) 05:42, 19 January 2006 . . Linas (Talk | contribs) . . 900×900 (71,319 bytes) (The winding number for the circle map. Original artwork by Linas Vepstas User:Linas <linas@linas.org>)