Fourier series in escape time fractals

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Offline v

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« on: December 19, 2018, 07:32:09 PM »
Taking the discrete fourier transform of the set of values associated with each point on the fractal set in the following manner:
\[
z_{n+1} = z_n^2 + c
 \]
then
\[
Z_k = \sum_{n=0}^{N-1} z_n \exp(-i 2 \pi k n / N)
 \]
where N is the set number of iterations. I've plotted |z_n| in black and |Z_k| in red for 3 points around and in the mandelbrot set. The points outside show divergence in the |z_n| plot and do not have a defined DFT, Z_k.  Points on the inside of the set appear periodic either because they are (in this example) or because the iteration is set too low to show divergence therefore have a defined Fourier series and DFT.

For formulas of other complex plane escape time fractals such as
\[ z_{n+1} = \exp(z_n) + c \]
or
\[ z_{n+1} = \sin(z_n) + c \]

a lot of the points are periodic with very high peaks and appear to diverge in escape time colour calculation.  One, more interesting, way of colouring these points is by assigning a colour to k that maximizes Z_k = DFT{z_n}(k) for n = 0 to number of iterations. The follow

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« Last Edit: December 21, 2018, 08:11:28 PM by v »


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