Image:Circle map poincare recurrence.jpeg

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[edit] Description

This figure shows the average Poincaré recurrence time for the iterated circle map modulo 1. 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)

Given a fixed value of K, Ω, θn and ε, the recurrence time N is given by the smallest integer N such that

|(\theta_{n+N}-\theta_n)\mod 1| < \epsilon\!

is satisfied. This figure shows the average value of N for a variety of starting values θn, and an ε = 0.003. Black indicates values of N of less than about 10, blue values of about 50, green shows values of about 140 and red shows values greater than 250.

The frequency Ω runs from 0 to 1 along the horizontal axis, and the coupling constant K runs from 0 at the bottom to 4π at the top of the image. The upper limit of the red area at the bottom occurs at K=1.

[edit] Summary

Average Poincaré recurrence time for the circle map. Original artwork created by Linas Vepstas User:Linas <linas@linas.org> 18 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
05:21, 19 January 2006 en:User:Linas (error in description)
04:41, 19 January 2006 en:User:Linas (<span class="autocomment"><a href="/wiki/Image:Circle_map_poincare_recurrence.jpeg#Description" title="Image:Circle map poincare recurrence.jpeg">→</a>Description -</span> Note)
04:35, 19 January 2006 en:User:Linas (<span class="autocomment"><a href="/wiki/Image:Circle_map_poincare_recurrence.jpeg#Description" title="Image:Circle map poincare recurrence.jpeg">→</a>Description</span>)
04:25, 19 January 2006 en:User:Linas (start adding sdescription)
04:14, 19 January 2006 en:User:Linas (Histogram of <a href="/wiki/Poincar%C3%A9_recurrence" title="Poincaré recurrence">Poincaré recurrence</a> for the <a href="/wiki/Circle_map" title="Circle map">circle map</a>. Original artwork by Linas Vepstas <a href="/wiki/User:Linas" title="User:Linas">User:Linas</a> <linas@linas.org>)

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