How to track vortexes in Julia set as you vary C

Question

Consider Julia set generated by iterating complex formula $z_{i+1} = z_i^2 + C $

typically there are some vortexes (centers of spiral patterns). e.g. this for $C= -0.512511498387847167 + i 0.521295573094847167$

https://en.wikipedia.org/wiki/Julia_set#/media/File:Julia_set,_plotted_with_Matplotlib.svg

Now when you smoothly vary C the position of these vortexes are moving

(see https://en.wikipedia.org/wiki/Julia_set#/media/File:JSr07885.gif).

can you think of any way how to track position of the vortex cores as you change C ?

Ideally if there would be some analytical method which does not require to evaluate numerically the value of Julias set at large number many points.

This is rather practical question, aimed to aplication in computer graphics, rather than pure math, I hope you would not consider it irelevant for Math stack-exchange

Answer 1

Let $$f_c(z) = z^2 + c$$ and iterate it like $$f_c^0(z) = z \\ f_c^{n+1}(z) = f_c(f_c^n(z))$$.

Now there are two fixed points $z_\alpha, z_\beta$ where $z = f_c(z)$, one is at the center of the left-hand biggest spiral and the other is at the right-hand tip. The other spirals and tips are pre-images of these fixed points, i.e. points $z_m$ where $$f_c^m(z_m) = f_c^{m+1}(z_m)$$ There are $2^{m+1}$ of these points, as each iteration of $f_c$ doubles the number of pre-images because $$f_c^{-1}(z) = \pm\sqrt{z - c}$$ has two values.

Practically, to find a $z_m$, you can use Newton's root finding method in one complex variable, given a sufficiently good initial estimate:

$$z_m^{(k+1)} = z_m^{(k)} - \frac{f_c^m(z_m^{(k)}) - f_c^{m+1}(z_m^{(k)})}{{f_c^m}'(z_m^{(k)}) - {f_c^{m+1}}'(z_m^{(k)})}$$

The derivatives can be evaluated iteratively: $${f_c^0}' = 1 \\{f_c^{m+1}}'(z) = 2 {f_c^{m}}'(z) f_c^{m}(z)$$ When implemented in code, be sure not to overwrite old values that are still needed for updating other variables.

To track the position of a given vortex, you would apply Newton's method for the next frame starting from the previous frame's position. This will only work reliably if the vortex does not move too much between frames, perhaps intermediate subframes will be necessary.