INTRODUCTION

The aim of this hands-on was to study a dynamical system. I have choosen to deal with a particular class of these systems which are the Iterative Function System (IFS). These systems are not really what one could call a "pure" dynamical system. Actually, all the systems we have studied so far can be written X'=F(X,t), that is to say that the derivative of a given set of variables X=(x1,...,xn) can be expressed as a function of this set X and of the time (for the non-autonomous systems). Thus, the whole evolution of the system is perfectly deterministic and actually, is totally determined by the initial condition X(0). However, it is well known that in many cases, the evolution of a system can become chaotic and then, the sensibility to the initial conditions can give the impression that the system follows a random trajectory. But one should still remember that this is just an illusion which is the consequence of our finite precision knowing of the initial condition.

The main difference between these "classical" systems and the IFS is that for the IFS, there is really a random parameter in the evolution of the system: at each time step (IFS are discrete systems), the next iterate is determined by applying a function choosen randomly amongst a set of pre-defined functions. To be more precise, the general algorithm of an IFS is given below:

• Choose M matrixes Ai in Rn*Rn, with i from 1 to M.
• Choose M vectors Bi in Rn, with i from 1 to M.
• Assign to each set (Ai,Bi) a probability pi.
• Choose an initial condtion X(0), which is also a vector of Rn.
• For k from 0 to Niter, do:
• Choose a random number i between 1 and M, so that Prob(i)=pi.
• Compute X(k+1)=Ai*X(k)+Bi.
• Then display all the X(k).
The set that is finally displayed is the trajectory of the system.

The fundamental difference with all the other dynamical systems is that chance is an explicit part of the algorithm: at each iteration, the choice of the function that is applied to X(k) is chosen randomly, according to the probability of this function.

This can seem to be completely absurd and one can say that such a behaviour never happends naturally.
This is another debate, but nevertheless, the IFS have been studied for about 10 years and nowadays, very powerful algorithms for image compression have been developped using this particular technique.

For this study, I choose to restrict my field of research to a particular class of IFS, which is still quite general:

• The iterates X(n) are complex numbers of the form X=r*exp(ja): this is a 2D system whose variables are "r" and "a".
• The transformations that I choose to deal with are only homothetias and rotations centered on Bi.
Finaly, intoducing ri the ratio of the homothetia and ai the angle of the rotation, this can be written:

(X(k+1)-Bi)=Ci*(X(k)-Bi)
with Ci=ri*exp(jai)
and i choosen randomly with a probability pi.

The next picture describes more clearly what happens at each iteration: This is all you need to know to discover The Fabulous World of IFS....  