Ufattr pictures

Attractors

The Mandelbrot set and other well known fractals are obtained iterating a formula of the type:

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

These are called discrete planar dynamical systems.It isn't very difficult to discover new formulas. You just invent a new one with some parameters (i.e. undefined constants) and try with different starting points and parameter values until you get a formula that gives you an image inside a window(this is an attractor). I have automated this procedure by letting the computer find these formulas with a program of mine called ufattr.In this way I have created a data base of around 50,000 formulas. This first picture

shows the kind of attractors that you get if you don't require them to be completely contained in the chosen window(I use the window -4<x,y>4).
Instead by requiring them to be always inside this window you get these kind of images,which are a lot more interesting.

The expert will recognize periodic and chaotic attractors. The periodic attractors are usually closed curves or spirals. The four images which follows are instead all chaotic attractors with color representing how many times the attractor has visited the pixel.

Route to chaos and origin of symmetry

The representation of the deformation step by step applied to a regular checkerboard allows the understanding of how a chaotic and/or symmetric attractor develops:

The three images show the first three iterations of a formula. You see that the final attractor (represented in blue) is starting to develop. In fact it is the final result (limit set in scientific terms) after many,many iterations.
You can thing of this process as a dough kneading process. The formula (like the baker) at each iteration will stretch the dough and then fold it back onto itself. When you do this(also if the dough is planar) you are working in a 3D space. You can think about doing this operations with deformations which are symmetric in space and that produce symmetric attractors in the plane. The following two pictures show two chaotic attractors with symmetric behaviour. They are attractors which are both chaotic and symmetric.