Image:Circle map bifurcation.jpeg

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Circle map showing mode-locked regions or Arnold tongues in black. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 4π at the top.
Circle map showing mode-locked regions or Arnold tongues in black. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 4π at the top.

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[edit] Description

The image shows the en:bifurcation diagram for the iterated en:circle map. The circle map is given by iterating on the map

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

This figure shows \theta_n \mod 1 along the x-axis, with coupling constant K ranging from 0 at the bottom to 4π at the top. The frequency Ω is held constant at 1/3.

Chaotic regions are interspersed with stable region are clearly visible. By comparing this image to the image showing Arnold tongues, and aligning along the Ω=1/3 slice, it becomes apparent that the stable regions correspond to the Arnold tongues, that is, to the regions where the map is mode-locked.

[edit] Summary

en:Bifurcation diagram for the en:circle map at Ω=1/3. Original image created by Linas Vepstas en:User:Linas <linas@linas.org> on 19 January 2006.

[edit] Licensing

GNU head 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".

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date/time username edit summary
05:00, 20 January 2006 en:User:Linas (<span class="autocomment"><a href="/wiki/Image:Circle_map_bifurcation.jpeg#Summary" title="Image:Circle map bifurcation.jpeg">→</a>Summary -</span> in-depth explanation of image)
04:44, 20 January 2006 en:User:Linas (<a href="/wiki/Bifurcation_diagram" title="Bifurcation diagram">Bifurcation diagram</a> for the <a href="/wiki/Circle_map" title="Circle map">circle map</a> at &Omega;=1/3. Original image created by Linas Vepstas <a href="/wiki/User:Linas" title="User:Linas">User:Linas</a> <linas@linas.org>)

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