Complex squaring map

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

Choose any complex number on the unit circle whose argument (complex angle) is not a rational fraction of π,
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 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 (radians) 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 z₀ has been chosen so that its argument is not a rational multiple of π, the forward orbit of z_n cannot repeat itself and become periodic.

More formally, the iteration can be written as:

$\qquad z_{{n+1}}=z_{n}^{2}$

where $z_{n}$ is the resulting sequence of complex numbers obtained by iterating the steps above, and $z_{0}$ represents the initial starting number. We can solve this iteration exactly:

$\qquad z_{n}=z_{0}^{{2^{n}}}$

Starting with angle θ, we can write the initial term as $z_{0}=\exp(i\theta )$ so that $z_{n}=\exp(i2^{n}\theta )$ . This makes the successive doubling of the angle clear. (This is equivalent to the relation $z_{n}=\cos(2^{n}\theta )+i\sin(2^{n}\theta )$ .)

Generalisations

This map is a special case of the complex quadratic map, which has exact solutions for many special cases.^[1] The complex map obtained by raising the previous number to any natural number power $z_{{n+1}}=z_{n}^{p}$ is also exactly solvable as $z_{n}=z_{0}^{{p^{n}}}$ . In the case p = 2, the dynamics can be mapped to the dyadic transformation, as described above, but for p > 2, we obtain a shift map in the number base p. For example, p = 10 is a decimal shift map.

References

↑ M. Little, D. Heesch (2004), Chaotic root-finding for a small class of polynomials, Journal of Difference Equations and Applications, 10(11):949–953.

v t e Chaos theory

Chaos theory	Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences

Chaotic maps (list)	Arnold tongue Arnold's cat map Baker's map Chua's circuit Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Duffing equation Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map

Chaos systems	Bouncing ball dynamics Economic bubble Tilt-A-Whirl

Chaos theorists	Michael Berry Leon O. Chua Mitchell Feigenbaum Martin Gutzwiller Brosl Hasslacher Michel Hénon Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Henri Poincaré Otto Rössler David Ruelle Oleksandr Mykolaiovych Sharkovsky Floris Takens James A. Yorke

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Complex squaring map

Chaos and the complex squaring map

Generalisations

See also

References