Complex squaring map: Difference between revisions

Revision as of 03:29, 6 August 2009

Revision as of 03:30, 6 August 2009

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In mathematics, the '''complex squaring map''', a [[polynomial]] mapping of [[Quadratic function|degree two]], is a simple and accessible demonstration of [[chaos theory|chaos]] in dynamical systems. It can be constructed by performing the following steps:

: (1)# Choose any [[complex number]] on the [[unit circle]] whose [[Arg_(mathematics)|argument]] (complex angle) is not a rational fraction of π,

: (2)# Repeatedly square that number.

This repetition (iteration) produces a sequence of complex numbers that can be described by their complex angle alone. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. In fact, it can be shown that the sequence will be [[Chaos theory|chaotic]], i.e. it is sensitive to the detailed choice of starting angle.