Complex squaring map

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 (angle) is not a rational multiple of π, # Repeatedly square that number. This repetition (iteration) produces a sequence of complex numbers that can be described alone by their arguments. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle.

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 (angle) is not a rational multiple of π, Repeatedly square that number. This repetition (iteration) produces a sequence of complex numbers that can be described alone by their arguments. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle. 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: 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) by Euler's formula.) This map is a special case of the complex quadratic map, which has exact solutions for many special cases. 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==

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