Looping animation made by sequencing key frames in the bifurcation diagram of the logistic map x := a x (1 - x), which is conjugate to the Mandelbrot set's quadratic polynomial x := x^2 + c.
The left hand edge of each keyframe is the parabolic root of a hyperbolic component (using jargon related to the Mandelbrot set). This can be found by Newton's method in 2 variables, starting from a nucleus of the relevant period (the animation starts 1, 2, 4, 8). The nucleus can be found by Newton's method in 1 variable, starting from a guess coordinate found by tracing external rays.
The right hand edge of each keyframe is a tuned / renormalized copy of the tip of the Mandelbrot antenna (c = -2 or a = 4). I found these coordinates by tracing external rays in the Mandelbrot set and then mapping c to a.
The top and bottom edges of the keyframes were found by iterating at the right hand edge and finding the two x that are closest (but not equal to) the starting point (the critical point x = 0.5). I think (not sure) these are at iteration numbers P and 2P, where P is the period of the "owning" hyperbolic component.
Interpolation between keyframes was done using Poincaré half-plane geodesics: