Dynamical Systems:possible uses

What you do with this information? It depends on the user. For an artist for example, this scan will turn out interesting shapes and he will try to enlarge the representation, try a different one, zoom on it. With the tools that I have now he can explore how the image changes changing some constant (parameter animation). Or he can try to breed it with other dynamical system seeing what comes out. There are three different representations implemented for each dynamical system: one is the "global one" used for the ordering, the other represents the attractor in color a la Golubitsky and the third is the escape time set where you see the basin of attraction of the attractor. This third representation will produce usually the most stunning fractal images after some massaging of the parameters contained in the formula.

For dydactic purposes I have written a program that will recover the usual algebraic formula (in fact the formulas found by the computers are stored as sets of numbers representing algebraic operations and constants). Once you have recovered the algebraic formula for some interesting attractor you can try to understand how the formula produces the image and introduce interesting variations. As an example of this exercise there is a list of some of the simplest formulas found by my program.

What you do is a kind of "reverse engineering" and you can build now a data base of synthetized formulas to which you can apply the same tools seen before. Of course you can introduce in this data base the formulas that you already know with some parameters and play with them by animating the parameters or breeding them. Especially interesting is parameters animation since it shows the onset of chaos with beautiful animated images.

For research purposes you can try to classify the dynamical systems. This is done by building lists of pointers to the images. So you can have a set of dynamical systems with threefold,fourfold,etc symmetry. You can try to discover unexpected shapes that can be connected to new phenomena,etc. In general,it is hoped that through a process of visual thinking the researcher after it get used to the data base, can have a more clear idea of the data stored and discover interesting new phenomena. The normal statistical analysis should now proceed more quickly since we are no more searching in the dark.

Now in concrete how one can use these tools in a collaboration?

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Maintained by Giuseppe Zito : Giuseppe.Zito@cern.ch