Identifying unordered set of complex data as originating from a particular Mandelbrot process? Here is an old question which probably originated back in my high school days.

When iterating the Mandelbrot

$$z_{i+1} = {z_i}^2+c$$

we get a sequence of complex numbers. If we plot these in the complex plane, we can get lots of different looks:



Now if we do not have ordered sequence, like a "time series" of these $z_k$ : $\{z_k,z_{k+1},\cdots
,z_n\}$ but instead just a plot like this with a unordered set of positions, can we calculate "backwards" what the original $c$ is?
You are drawing critical orbit = forward orbit of a critical point.

You ask : can we read/approximate the parameter c from the graph/image of critical orbit ?

I think that it is possible to some extent:


first image: c is probably from interior of Mandelbrot set main cardioid (=  internal radius &lt; 1), with internal angle smaller then  n/8 ( 8 arms , turning clockwise), for example c = 0.345805039440922  +0.097755675960631 i 
last image looks like a periodic orbit with period = 14 , so it maybe center of hyperbolic component with period 14, like for example c = 0.295453572980359  +0.020451548527456 i   
middle image looks like Siegel disc


But remember that because of slow dynamics and numerical errors first n-points may look different then images with more points.

The next questions could be : 


how critical orbit is changing when c is moving on the parameter plane?
how to move on the parameter plane ? 


HTH
You are drawing critical orbit = forward orbit of a critical point.

You ask : can we read/approximate the parameter c from the graph/image of critical orbit ?

I think that it is possible to some extent:


first image: c is probably from interior of Mandelbrot set main cardioid (=  internal radius &lt; 1), with internal angle smaller then  n/8 ( 8 arms , turning clockwise), for example c = 0.345805039440922  +0.097755675960631 i 
last image looks like a periodic orbit with period = 14 , so it maybe center of hyperbolic component with period 14, like for example c = 0.295453572980359  +0.020451548527456 i   
middle image looks like Siegel disc


But remember that because of slow dynamics and numerical errors first n-points may look different then images with more points.

The next questions could be : 


how critical orbit is changing when c is moving on the parameter plane?
how to move on the parameter plane ? 


HTH