Global representation of dynamical systems

Before starting to read the explanation, perhaps you want to look at the global representation for 12 dynamical systems of the test data base.

Let's consider a dynamical system in a given window:

  -4 < x < +4
  -4 < y < +4

For each one of the (infinite) points in this square, you have a different orbit and a possibly different pattern. The problem is :how do we represent all these possible images with only one single image, that gives the global behaviour of the dynamical system?

The solution to this problem is based on the fact that symmetry,chaos and other properties in a dynamical system are not affected by the precision of the numbers used to carry out the iterations. So I limit my precision to the pixel size. If I have 700x700 pixels in the viewing window, I set up two matrices: INE(700,700) and JNE(700,700) and compute my iteration in the center of each pixel I,J only once, storing in INE(I,J) and JNE(I,J) the result of the computation in the form of address of the pixel where the formula moves the center of the starting pixel. When I have computed the iteration once for all pixels, I proceed to analyze the results.

Starting from each pixel and following the address of the next pixels in INE,JNE I can have three cases:

The orbit exits the viewing window. These pixels are coloured like we do with escape time representation,i.e. with color indicating the number of steps until the orbit exits the window.
The orbit returns to its starting point. These are the periodic limit sets all colored with the same color.
The orbit ends in a limit set. These also are colored with the same color.