What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which describe this approximation. For example, I know that the period 2 region boundary is C = -1 + 1/4*exp(I*t). For higher periods, exact solutions are not easily found, so a Taylor series is the closest one can get to an exact solution. I did find a paper http://www.ams.org/journals/proc/1995-123-12/S0002-9939-1995-1301497-3/S0002-9939-1995-1301497-3.pdf that gives the exact solution for period 3 bulbs. For period 4 the math gets too difficult.
Can someone point me to some pseudo-code or an algorithm that describes how to derive a Taylor series approximation to a period n hyperbolic component boundary? I've thought of a way to calculate a sampling of locations along the border, then do a least squares curve fit to the points. I am wondering if there is any other way.
Pseudo-code would be helpful in order to be able to write the program in the language of my choice. I have seen examples of code written in various languages, but it is very hard to decipher the algorithm used.
Other ways : solving system of boundary equations using Newton method
example 1
example 2
HTH