Transferred from en.wikipedia to Commons by Maksim.

Summary

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.

Summary

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

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. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
	You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
	This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

Original upload log

Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version.

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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>)