Amazing that it's not layered, as it really does look that way. Excellent work!
I, too, would also like to know more about Herman rings.
Like a Siegel disk, except hollow. The normal M-set can't produce them, nor any polynomial map of one complex variable, but some rational maps of one complex variable can, of which Supernova is one such map:
If
b = 0,
a is on the unit circle with irrational angle, and
c is pure real or pure imaginary and |
c| > 3, Herman rings tend to occur in the dynamic plane. The images aren't hugely fascinating compared to the parameter space images from the same map, though, particularly
c-plane images with suitable choices of
b and
a. There you find "bud mirror" lines on the axes and sometimes other "bud mirror" arcs elsewhere; the dynamic plane has Herman rings for irrational points (as defined ... somehow. "Not exactly at any bud-pair's base", I think) on these mirrors. Elsewhere it has up to four attractors, three finite and one superattracting unmoving one at infinity. The finite ones depend on the parameters and if
b = 0 one is stuck at zero and also superattracting, regardless of
a and
c. Images of all slices contain seahorse or other shapes that have been hollowed, with points going to infinity surrounding a lake of points going to zero or vice-versa; changing
b away from 0 moves the latter attractor off zero, makes it less stable (or even
unstable, bifurcating or simply ceasing to exist), and makes the "goes to zero" regions gnarl, distort, fragment, or convert into thick foam. Moving
b into a
b-plane minibrot
really does interesting things to those regions.
There are images, mostly parameter plane, here; search the gallery for images tagged "supernova" (all the ones by me, and probably some of the others) or go to these
big zoom sequences in parameter space some of which show some of the effects of moving
b off 0.
January 01, 2013, 07:42:59 PM
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