Anyway, in studying chaos theory I got sidetracked by these pretty little things. These 'things', like the one shown here, are chaotic attractors.
To understand what an attractor is I'll use an analogy with biochemistry. Think about protein folding. You've got an initially unfolded protein in some environment, ie. a buffer solution. As time goes on the protein will eventually fold into some stable conformation. The conformation it folds into could be called an attractor of the system since it 'attracts' the parameters defining the protein and causes them to settle into some final values.
So, an attractor is just a general way of representing the state of a given system, like your protein, after a long time. For example, if you've ever plucked a guitar string and watched it slowly stop vibrating, you've observed an attractor. The final position of the string, really just a single point, is the attractor.
For the case of the guitar string the attractor is just one point. No matter how you pluck it the string will, after time, stop moving. You can predict its future absolutely. A chaotic attractor, on the other hand, has no such absolutes. The system it describes does settle to final set of states, it's just not possible to say which state. The diagram above is an example of such a chaotic attractor.
Sorry for the lengthy introduction to chaotic attractors. I think I'll be posting one for the quadratic map (like the one above) every week. (And maybe other systems too...) I'll include a calculation of the largest lyapunov exponent, a measure of how chaotic it is, and the dimension too. Just for fun. Of course, you don't have to care. Just look at it and appreciate how pretty it is. :)
D = fractal dimension = 1.6
L = largest lyapunov exponent = 0.329
