Complex squaring map: Difference between revisions

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

# Choose any [[complex number]] on the [[unit circle]] whose [[Arg_(mathematics)Complex_number#Polar_complex_plane|argument]] (complex angle) is not a [[rational fractionnumber|rational]] multiple of π,

# Repeatedly square that number.

This repetition (iteration) produces a [[sequence]] of complex numbers that can be described alone by their complex angle alonearguments. 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, itIt 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 (a) the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but (b)angles thewhich anglediffer mustby bemultiples limited to lie on the circle, i.e. the range zero toof 2π ([[radiansradian]]s) are identical. Thus, when the angle exceeds 2π, it must ''wrap'' to the remainder on division by 2π. NowTherefore, [[pi|π]]the angle is atransformed according to the [[transcendentaldyadic numbertransformation]] and(also theknown fractionalas partthe has2''x'' nomod known1 patternmap). Thus, even ifAs the initial anglevalue is chosen such that it''z''₀ has abeen chosen finiteso numberthat ofits digits andargument is knownnot witha completerational accuracy, the remainder on division by 2π inherits the lackmultiple of pattern inπ, the fractional[[Orbit part(dynamics)|forward of π. The iteration therefore "reads out" the patternless detailorbit]] of the''z''_''n'' fractionalcannot partrepeat of π,itself and this is in fact the origin of the complexity of the anglebecome sequenceperiodic.

More formally, the iteration can be written as:

:<math> \qquad z_{n+1} = z_n^2 </math>

where <math>z_n</math> is the resulting sequence of complex numbers obtained by iterating the steps above, and <math>z_0</math> represents the initial starting number. We can solve this iteration exactly:

:<math> \qquad z_n = z_0^{2^n} </math>

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,(This is equivalent to 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 theby [[logistic map]], for example, and leads to exponential divergence of close starting angles (see [[Lyapunov exponent]]Euler'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 mapformula]].)

== Generalisations ==

This map is a special case of the complex [[complex quadratic map]], which has exact solutions for many special cases.<ref>M. Little, D. Heesch (2004), [http://www.engmaxlittle.ox.ac.uk/samp/membersnet/publications/GDEA41040.pdf Chaotic root-finding for a small class of polynomials], ''Journal of Difference Equations and Applications'', '''10'''(11):949&ndash;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 ''p''&nbsp;=&nbsp;2, the dynamics can be mapped to the binarydyadic [[2x mod 1 map|shift map]]transformation, as described above, but for ''p''&nbsp;>&nbsp;2, we obtain a shift map in the [[number base]]&nbsp;''p''. For example, ''p''&nbsp;=&nbsp;10 is a decimal shift map.

== See also ==

* [[Logistic functionmap]]

* [[ChaosDyadic theorytransformation]]

* [[Logistic function]]

* [[List of chaotic maps]]

* [[Complex quadratic map]]

==References==

{{reflist|2}}

{{wikibooks|Fractals/Iterations_in_the_complex_plane/q-iterations#Dynamic_plane_for_c.3D0}}

{{Chaos theory}}

[[Category:Chaotic maps]]