Strange Attractors

A note on this project: I started investigating these after viewing Paul Bourke's page on random Strange Attractors. I fell in love with the fractals that he had and I needed to have my own :] So I used his generator (which is posted on his page) and modifided it significantly so that it would generate 3D attractors instead of 2D ones. The results I got were very appealing.
In this page I am going to display some of the better results I have obtained, and attempt to give a few explanations as to how these buggers work.

Galleries:

Explanation:

Basics of sequences:
A sequence in the reals a_n is a list of numbers which depend on a natural number n. (from the absolute basics) A sequence may either converge to a number, diverge to infinity, alternate between fixed values, or simply be chaotic. Here are some simple examples: a_n = 1 + 1/n converges to the number 1.
a_n = n diverges to infinity.
a_n = (-1)ⁿ alternates between -1 and 1.
a_n = sin(n) behaves "chaotically". A recursive sequence is a sequence where a_n may depend on previous members of the sequence (where some initial conditions are defined): a_n = a_n-1 + n^-2. With a₀=0 converges to (interestingly enough) π²/6
a_n = a_n-1 + a_n-2. With a₁=1 and a₂=1 is the Fibonacci sequence, but for our purposes it simply diverges to infinity.
Sequences may not necessarily be only in the reals, either: a_n may be a 3-tuple of (x,y,z) and then we say a converges when x, y, and z converge. It diverges to infinity if either x, y, or z fly off to infinity.

Attractor sequences:
This covered we can get into the juicy details:
Typically in Calculus or Analysis classes, the problem is to find of if a sequence is convergent or divergent. Here we are concerned mainly with sequences which do not converge, are not periodic, and do not diverge to infinity either. These are called strange attractors. In order to generate these, it is necessary to use a recursive sequence which is at least quadratic (linear recurrences are not interesting ;] ). This means that x_n, y_n, and z_n depend on x_n-1² and other values with two components. This is hard to explain, but the formula is much easier to understand.

Or, in Matrix form:

Where A is a 3x10 matrix.

When referring to the configuration of coefficients for an attractor, I will simply refer to it as A, meaning the 3x10 matrix which acts on the above.

It is also possible to use cubic attractors, which have up to three components for the expressions, but I'm not going to write that out here. The matrix for those is 3x20, meaning that it has 60 elements. This is quite a lot when it comes to chaos and random possibilities.

I made these attractors by generating each element of A randomly, and testing against several common conditions: namely the attractor shooting off to infinity, or converging to a fixed point. Also, it calculates the Lyapunov exponent which essentially measures the chaoticness of the attractor. I'm still shaky on the understanding of it, but if you wish to understand more, visit Paul Bourke's page on attractors. The Lyapunov exponent is negative when the attractor is periodic, positive when it is chaotic, and around zero when the attractor is relatively convergent. The program filters out only the attractors with positive Lyapunov exponents, and saves a whole slew of them in POV-Ray format, and then I tediously sort through and pick out the interesting ones.

Discoveries:
When I began examining the patterns that formed out of generating the entries of A randomly (between -1 and 1). I didn't notice right away (but should have) a pattern that emerged in the behavior of randomly selected entries. By far, most of them diverge to infinity. The second largest group is those which are stable, and then the ones which are periodic, the number of occurences of chaotic attractors is the least of them all, generally there is only about 1 per 1000 tests. Consider for instance, picking random points in the complex plane and testing to see whether it corresponds to a Julia set. The probablity of picking one is much more likely than picking a chaotic attractor, but what about picking a Julia set of a 30 dimensional Mandelbrot? Much more rare.

I am guessing that the entire space of strange attractors cooresponds to one MASSIVE 30 dimensional fractal. The one snag in the analogy to Julia sets, though, is the fact that Julia sets are periodic. (When you find a point that is actually in the set, and iterate it, it will land always in some point that is also in the set, and furthermore, actually "converge" to some finite permutation of points, but that's beside the point.) Strange attractors also have some configurations where an attractor "converges" to a loop in 3D space. These are difficult to explain, and I know of no nice analogy to those either.

I became aware of these patterns when I began looking at animations, and comparing how moving from one chaotic attractor to another via a straight path can jump to infinity and to stability rather quickly, much like what happens when moving from one Julia set to another.

Conclusion:
A proposition that I have come to is that the set of *stable* attractors is a fractal. (And a 30 dimensional one, at that.) I have made a few cross sections of this creature (in one of the galleries above), which may be insightful if you are interested in understanding the way these work. How to interpret, much less visualize such a fractal remains a mystery to me, but hopefully I'll find some interesting method some day.

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To get in touch with me, my email is coa@andrew.cmu.edu. No death threats, please.