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Author Topic: Inside  (Read 1030 times)
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reesej2
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« on: April 22, 2010, 11:32:59 PM »

I've been thinking about the complexity inherent in the structure outside the Mandelbrot, and noticing that the interior doesn't seem to have nearly that detail no matter what means are used to color it. I'd be interested to know if anyone knows of any strategies that get that kind of detail...

Pickover stalks and similar start to get close, but it's still a very simple pattern. Distance estimation does practically nothing, and escape time is of course trivial. Any ideas?
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teamfresh
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fractalco
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« Reply #1 on: April 23, 2010, 04:05:50 AM »

this really should be looked into!
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hobold
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« Reply #2 on: April 23, 2010, 03:55:09 PM »

Well, the points inside the Mandelbrot set certainly go through interesting and varied orbits. Even those in the main cardioid, that ultimately converge. Furthermore, the orbits vary smoothly (except at the individual points where different cardioids/disks touch each other).

The hard part lies in "distilling" some invariant simple numbers from those orbits, that can be mapped to a colour palette, but capture enough information from the orbit to be actually interesting.
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lkmitch
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« Reply #3 on: April 23, 2010, 05:59:05 PM »

I think the key is to look at inside points differently.  For interior points (as opposed to boundary points), you may wish to begin by plotting the orbits of some target points to get a feel for the types of orbits.  Generally, the closer a point is to the center of its respective disk or cardioid, the more quickly its orbit will converge to a limit cycle (thus, potentially less interesting graphically).  I like to think about questions about the orbits that can be answered graphically.  For example, how often (percent of iterations) is the real part positive?  How close does the orbit come to a particular point or shape?  If the orbit is chaotic, how chaotic is it?  If I connect the iterates, what kind of pattern will it make?  If the orbit is ultimately chaotic with period n, what happens if I draw lines between every (n-1) or (n+1) points?

True, you won't find the same type of detailed structure with inside points, but there is structure there nonetheless.
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reesej2
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« Reply #4 on: April 23, 2010, 09:10:19 PM »

Oh, yes, certainly there IS structure there--that can be seen by simply plotting Pickover stalks or preperiod iterations. The thing is that the structures produced that way are not nearly as rich as the exterior, which seems strange, since the paths of the interior points are at least as varied as those of the exterior.

I think connecting the iterations has been done, I read a paper on it a while ago--you tend to get a sort of blur, not very exciting. But the idea of connecting points at intervals based on the period is interesting...

Also, on the notion of "how chaotic is it", perhaps plotting the Lyapunov exponent would be relevant? And maybe the Lyapunov exponent of the nth-iterate map, for various n?
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