Bubble Intersection

This video illustrates the Mandelbrot set of the fourth degree mappings f(x) = x^2((x+b)/(cx+1))((x+25)/(25x+1)), with complex values b and c.  The value of c follows the circle of radius 1 centered at the origin in the complex plane.  The video images show the plane of values of b, with the value of b at the center of each image equal to the complex conjugate of c.  Note that there are two curves (with bubbles on both sides) that cross at the center of the images.  These curves indicate the presence of Herman rings in the Julia sets of f(x) when b lies along these curves, and the two Herman rings coincide when b is at the image center, where the two curves cross each other.  A point b in the image is colored based on the result of iterating the four critical points of f(x).  If all the critical points are attracted into periodic cycles, the color of b depends on the largest period.  If any critical point lasts 100000 iterations without falling into a periodic point, then b is colored white.  The point is colored green if the critical points iterate to infinity, or blue if all critical points converge to zero.