(Question) What is this phenomenon called?

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Offline AlexH

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« on: October 23, 2018, 03:17:48 AM »
I make a lot of Julia set animations. I usually use an ellipse to map out what the value of C will be on any given frame (see the attached scribble diagram). Normally I'd expect that when you start at a certain C value, all features of a fractal should return to their starting positions by the end of the animation as long as you return the value of C to its initial value at the end of the animation. But that's not always the case. I've noticed that with certain Julia sets (it might be all of them, I don't know), if you plot that ellipse around a certain point (it changes based on the equation), the animation behaves completely differently. Instead of the features returning to their starting positions after just one loop, they take multiple loops to return to their starting position. The periodic movement of the the features goes completely haywire and can even change based on what iteration the feature is attached to.

If none of that made sense, here are a couple of examples. The left animations have the same number of frames as their counterpart on the right. The left ones take multiple loops to return all of their features to their original position, but the ones on the right only take one loop.





So my questions are:
1. What is this called, or what terminology would you use to describe it?
2. Why does this occur?
3. Can you use the fractal's equation to predict where these points are?

Offline gerrit

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« Reply #1 on: October 23, 2018, 03:40:05 AM »
As a Julia set is uniquely defined by c it must be a bug. Perhaps your angle increment is such that you don't return exactly to the starting value?

Offline AlexH

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« Reply #2 on: October 23, 2018, 04:05:38 AM »
As a Julia set is uniquely defined by c it must be a bug. Perhaps your angle increment is such that you don't return exactly to the starting value?
It's not a bug. Sorry, I didn't do a great job explaining it, haha.
The image that the fractal starts on is the same at the beginning and end of the loop, but if you trace any individual feature with your finger, you'll notice that it takes multiple loops for it to return to its starting point.

Take bottom animations for example. They're both 50 frames at 30 fps, so they take about 1.66 seconds to play a single loop. The value of C on the left orbits around 0+0i, the other one doesn't (it's some point in quadrant 2). I've included a C Map showing the difference between the two. The big circle is where the C values of the animation with the phenomenon are being pulled from (the left animation) and the little circle is approximately where the C values of the animation on the right are being pulled from.
In the animation on the right, all of its features return to where they start. In the animation on the left, the features get passed around from loop to loop. The smaller features with higher iteration counts tend to take even longer than their lower-iteration cousins. If you keep zooming in and keep increasing the iteration count, the amount of time for any given feature to return to its starting position just keeps increasing forever.


*edit*
Here's a zoom in on the left animation. This animation is also 50 frames long, so it's the same as the one above. Look how many cycles it takes for a green feature to return to its starting position versus the larger magenta ones. Even though the C value uniquely defines a Julia set, moving the C values around 0+0i doesn't return everything to its starting point in the way that you'd expect.

« Last Edit: October 23, 2018, 04:20:09 AM by AlexH »

Offline gerrit

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« Reply #3 on: October 23, 2018, 04:42:44 AM »
I get it now. Not sure about explanation, playing around with the idea on some polynomial M-set like fractals I can see it sometimes happening if the Julia c loop goes around a disconnected island/mini.

Offline pauldelbrot

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« Reply #4 on: October 23, 2018, 05:31:35 AM »
Such a loop induces an automorphism on the Julia set, but not always the identity automorphism. There must be special points in the parameter plane, likely zeros and poles of some function, where this occurs and the winding number to put things back is greater than one.

The position of an affected feature, in turn, must involve an nth root. The purple orbs' centers would have to be decided by square roots of c. Now one complete loop that misses the center just wiggles them around but a loop that encloses 0 exchanges the positive and negative square roots, since square root is a two-ply function. And this happens because the origin is a zero of the square root function.

There's a connection here also to path integrals on the complex plane. There's a function that determines how a feature moves as a function of c moving, essentially the above square root involving function's derivative with respect to c. If you move c along a short arc, the movement of the feature integrates that function over that arc. If the arc is a complete circle, then you've integrated the function around a complete circle. If a function of complex numbers is continuous, analytic, and lacking zeros and poles on such a circle and its interior, the path integral is zero, but it can be integer multiples of some nonzero value when the closed path of the path integral encloses a pole, at the very least. The square root is vertical at 0 so the derivative of this function indeed has a pole there.


Offline AlexH

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« Reply #6 on: October 24, 2018, 05:56:26 AM »
Thanks for the replies. It'll probably take some time for me to fully digest what you've told me (I kind've bumbled my way into this hobby with only a Calculus II background, haha), but I do understand some of the language you've used. Although two-ply is a term I normally hear applied to toilet paper. XD


Just for fun, here are the equations I used:
Black and White animation.
The animation orbits around \( -0.715+0.7141428571428571*i \):
\[ Z_{n+1}=\frac{1-{Z_{n}}^{2}+C^{16}}{2+4*Z_{n}}+C \]

Color animation.
The animation orbits around \( 0+0*i \):
\[ Z_{n+1}=\frac{1+{Z_{n}}^2}{1-{Z_{n}}^2}+C \]

Offline fractower

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« Reply #7 on: October 25, 2018, 06:09:22 PM »
How about calling the phenomenon SuperCycle = apparent period multiplication caused by some underlying symmetry.


Offline claude

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« Reply #8 on: October 26, 2018, 07:43:11 PM »


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