I am trying to find a way to calculate the Lyapunov exponents of the Mandelbrot set. There are some very nice diagrams that you can find on Flickr of a plot of the Lyapunov exponents of the Mandelbrot set, leading to a much higher detail in the structure of the set.
I would like to do this as well and have found the definition of the Lyapunov Exponent but I cannot find a good paper or documentation on how to calculate the Lyapunov Exponent for the M-Set. Especially the derivative of the function $z[n+1] = z^n + c$ with regards to the starting position $x_0$ is what I cannot figure out.
Do I have to use an approximation for this derivative? If yes, which one? If no, how can these exponents be calculated for the Mandelbrot set.
Many thanks!
Lukas M.
PS: I am a hobby mathematician, but mainly an engineer. A description in terms of an algorithm would greatly help. Thank you for your consideration.
$$ \lambda_f(z_0) = \lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=0}^{n-1} \left ( \ln\left|f'(z_i)\right | \right ) $$
where :
$$ f'(x) = \frac{d}{dz}f_c(z) = 2z $$ means first derivative of f with respect to z
see : https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set for references