Evolving Inner and Outer Rings

The music in this video is “Whiny Funk in E Major” by Jack Lamos.

This video animates Julia sets of the fourth degree rational functions
f(z) = x^2(z+b)(z+c)/(bz+1)(20z+1),
following a curve through the space of complex pairs (b,c). Most of the Julia sets in the video have two different Herman rings: one ring is inside the unit circle and a larger ring encircles the inner ring.  Points that iterate into the inner ring are colored green, and those iterating to the outer ring are colored blue.

The values of b are near the circle of radius 1, with argument decreasing during the animation from about 170 degrees to about 16 degrees.  The values of c are real, very close to 20.
The winding numbers of the inner ring decrease from near .5 toward 0 during the animation, while the winding numbers of the outer ring increase from .5 toward 1.

The values of b and c in these images are calculated using the Mandelbrot sets shown in the video “Rotating Double Rings” on this channel.  That video illustrates two curves of Herman rings for the function f(z) when |b| = 1.  Increasing or decreasing the value of |b| causes the two curves to expand or contract relative to one another.  For each frame of this animation (i.e., each value of arg(b)), a value of |b| is calculated so that the two curves intersect.  Where those two curves intersect, the corresponding Julia set has two different Herman rings.  

In these images, points that converge to 0 are colored yellow, points converging to infinity are colored pink.  Points that converge to the inner Herman ring (or a nearby periodic cycle) are colored green.  Points that converge to the outer Herman ring (or a nearby periodic cycle) are colored blue.  At the start of the video we see the entire Julia set.  We zoom in, approaching the origin, centering on the inner ring at about 33 seconds.  Toward the end, we zoom out to illustrate the connected component of the inner ring as its winding number decreases to about 0.16.