Complex squaring map: Difference between revisions

Revision as of 11:31, 7 June 2012

Revision as of 19:46, 10 November 2012

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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.

== Chaos and the complex squaring map ==

The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π ([[radiansradian]]s) are identical. Thus, when the angle exceeds 2π, it must ''wrap'' to the remainder on division by 2π. Therefore the angle is transformed according to the [[dyadic transformation]] (also known as the 2x mod 1 map). As the initial value ''z''₀ has been chosen so that its argument is not a rational multiple of π, the [[Orbit (dynamics)|forward orbit]] of ''z''_''n'' cannot repeat itself and become periodic.

More formally, the iteration can be written as: