How to draw these circles? I'm looking for a method which I can do it with Adobe Illustrator. (or draw in GeoGebra and export to illustrator)
The lighter circles seem to have centers that lie on a common circle $S$, and have radii twice that of $S$.
In GeoGebra, try
and fiddle with the details. The thicker circle that is concentric with the whole picture seems to have the same radius as the others, so you can tack on a
The darker spirals are arcs of the circles that the code for the light circles produces. So try