Image:Devils-staircase.svg

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Devils-staircase.svg‎ (11KB, MIME type: image/svg+xml)

[edit] Description

This figure shows the winding number for the circle map, as a function of Ω, with K held constant, at K=1. It is an example of the Devil's staircase. Each of the flat regions corresponds to a region of phase-locking; that is, each flat area is a slice through one of the Arnold tongues.

The circle map is a model of the phase-locked loop, and is given by iterating the equation

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

The winding number is defined as the limiting behavior of the system after many iterations, by:

$\omega=\lim_{n\to\infty} \frac{\theta_n}{n}.$

This graph shows ω on the vertical axis as a function of Ω on the horizontal axis.

[edit] Image history

Original image created by Linas Vepstas User:Linas on 17 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 Devils-staircase.svg.
Images which have been tagged with this template may be deleted immediately after satisfying these conditions (CSD I8).

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(del) (cur) 15:15, 17 January 2006 . . Linas (Talk | contribs) . . 600×480 (10,737 bytes) (Original image created by Linas Vepstas User:Linas 17 January 2006)