Start with a Cartesian point $(x,y) \in \mathbb{R}^2$, convert it to polar coordinates $(r,\phi)$ ($\phi$ in radians), and then reinterpret $(r,\phi)$ as $(x,y)$, i.e., set $$(x,y) = (r,\phi) \;.$$ And now repeat. Here is a typical example of the results. The red points represent starting $(x,y)$ points (random within $[\pm 3]^2$), and the blue lines track (in this case) $8$ iterations:
It forms a curious and attractive pattern deserving of explanation. Clearly there is a "circling" around $(1,0)$, but the circling is not quite circular.
Q1. What is a succinct explanation of the overall pattern?
Q2. What is the limiting shape of the vortex surrounding $(1,0)$?