Complex squaring map: Difference between revisions

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Revision as of 09:37, 5 August 2009

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Starting with angle θ, we can write the initial term as <math>z_0 = {\exp}(i\theta)</math> so that <math>z_n = {\exp}(i2^n\theta)</math>. This makes the successive doubling of the angle clear. Also, the relation <math>z_n = {\cos}(2^n\theta)+i{\sin}(2^n\theta)</math> makes the wrapping of the angle onto the circle obvious. The doubling-and-wrapping is the same origin of chaos as the stretching-and-folding of the [[logistic map]], for example, and leads to exponential divergence of close starting angles (see [[Lyapunov exponent]]s). If the initial angle is expanded in a binary representation, then the iteration can be clearly seen to be equivalent to the [[2x_mod_1_map|shift map]].

== Generalisations ==

This map is a special case of the complex [[quadratic map]], which ishas solvableexact insolutions for somemany special cases.<ref>M. Little, D. Heesch (2004), [http://www.eng.ox.ac.uk/samp/members/publications/GDEA41040.pdf Chaotic root-finding for a small class of polynomials], ''Journal of Difference Equations and Applications'', '''10'''(11):949-953.</ref> The complex map obtained by raising the previous number to any natural number power <math>z_{n+1} = z_n^p </math> is also exactly solvable as <math>z_n = z_0^{p^n}</math>. In the case <math>p=2</math>, the dynamics can be mapped to the binary [[2x_mod_1_map|shift map]], as described above, but for <math>p>2</math>, we obtain a shift map in the [[number base]] <math>p</math>. For example, <math>p=10</math> is a decimal shift map.

== See also ==

* [[Chaos theory]]

* [[List of chaotic maps]]

* [[Logistic function]]

* [[List of chaotic maps]]

* [[Lyapunov stability]] for iterated systems

* [[Complex quadratic map]]