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.

[edit] Description

The image shows the bifurcation diagram for the iterated 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

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

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(del) (cur) 05:05, 20 January 2006 . . Linas (Talk | contribs) . . 450×900 (120,433 bytes) (Reverted to earlier revision)
(del) (rev) 05:04, 20 January 2006 . . Linas (Talk | contribs) . . 450×900 (120,130 bytes) (Reverted to earlier revision)
(del) (rev) 05:03, 20 January 2006 . . Linas (Talk | contribs) . . 450×900 (120,433 bytes) (Bifurcation diagram for the circle map at Ω=1/3. Original image created by Linas Vepstas User:Linas <linas@linas.org> on 19 January 2006.)
(del) (rev) 04:44, 20 January 2006 . . Linas (Talk | contribs) . . 450×900 (120,130 bytes) (Bifurcation diagram for the circle map at Ω=1/3. Original image created by Linas Vepstas User:Linas <linas@linas.org>)