Strange Attractor

For a dynamical system the attractor is where it will end up eventually. This can be a fixed point like a ball bearing in a bowl. Or an orbit or cyclic attractors like a planet around the sun. Or we may have a system that never returns in the same place like the toy "cosmic balls" - this we call a strange attractor

To be more precise, if we consider the system's state space i.e. the list of all possible states for the system, then a strange or chaotic attractor is a set of states in a system's state space with very special properties. First of all, the set is an attracting set. So that the system, starting with its initial condition in the appropriate basin, eventually ends up in the set. Even if the system is perturbed off the attractor, it eventually returns. Second, and most important, once the system is on the attractor nearby states diverge from each other exponentially fast.

It is easy to generate strange attractors in the plane:just try a formula of the type:

   x = f(x,y)
   y = g(x,y)

with different values of constants and starting points until you get an image (orbit) that remains always in a chosen "window" for example -4<x,y>4 For example this image shows 30 attractors(not all strange) found in this way.

Chaotic attractors have geometric structures that are fixed and unchanging---despite the fact that the trajectories moving within them appear unpredictable. In this sense, the chaotic attractor's geometric shape is the order underlying the apparent chaos.

The geometric shape of a chaotic attractor implements a kind of dough kneading. The local separation of trajectories corresponds to stretching the dough and the global attraction property corresponds to folding the stretched dough back onto itself. One result of the stretch-and-fold aspect of chaotic attractors is that they are fractals---some cross-section of them reveals similar structure on all scales.

To understand this let's consider the following formula:

   x = x*y+a*x-y      with  a=0.357057        
   y = x+y
 

The attractor shape shown at the top of this document, reminds a seashell and is typical of attractors. You can better appreciate the elegance of the shape and its apparent 3D pattern in this color rendering of the same attractor.

To understand how this shape is created, we consider a red checkerboard 200x200 and deform the squares using the formula. This sequence of three images shows how the checkerboard is deformed in the three first applications of the formula. The attractor is superimposed and is the result of continuing this procedure many many times or, in scientific terms ,the limit set. But you can see that also after 3 passes, the final shape is starting to appear.