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, $θ n$ 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 $θ n$ , 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

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.

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(del) (cur) 04:14, 19 January 2006 . . Linas (Talk | contribs) . . 450×900 (132,528 bytes) (Histogram of Poincaré recurrence for the circle map. Original artwork by Linas Vepstas User:Linas <linas@linas.org>)