First steps:
1 complex function.






Before studying very complicated cases, the simplest way to really understand what happends in an IFS, is to take a look at the simplest system that can be found: an IFS with only one function.

Actually, as there is only one function, this particular case is a deterministic problem which is in fact very classical.

The equation of evolution of the system can then be simply written:

(X(n+1)-B)=r*exp(ja)*(X(n)-B)

and we can then conclude that:

(X(n)-B)=rn*exp(jna)*(X(0)-B)

The nth iterate is obtained by applying n times an homothetia of ratio r and a rotation of angle a, both centered on B.
We can clearly see that the system is totally deterministic, as the nth iterate only depends on the discrete time n, the parameter c=(r,a) and the initial condition.

It is now clearly obvious that three major behaviour can be distinguished:
 

Trajectory of a divergent system
Trajectory of a convergent system
We can already see that a main concept is appearing in these very simple cases: the existence of an attractor for the system. In fact, it is not obvious to give a clear definition to this concept at this point of the discussion and for now, we might just say that the set A is the attractor of a system if:
*    the modulus of X(n) remains finite when n becomes very large.
*    for any initial condition X(0) and when n becomes very large, the "distance" between X(n) and A becomes very small (the distance between a point X and a set A will be defined further, but is quite intuitive).

With this definition we can say that:

*    In the first case ( | r | > 1 ), the iterates become larger and larger in modulus when n increases and so, they clearly does not respect the first condition: the system does not have an attractor.

*     In the second case ( | r | < 1 ), the set A={B} is clearly the attractor of the system.

*    When | r | = 1, we must pay a little much more attention to determine if the system has an attractor:
+ In the first case, one may think that A=O. However, an orbit O explicitly depends on the initial condition X(0) and so forth cannot be considered as the attractor of the system.
+ In the second case, for any value of X(0), the iterates X(n) will travel on the whole circle of radius 1 centered on B. This circle respects the two conditions recquired above and thus is the attractor of the system.
 

After this brief overview of the most simple IFS we can already make a few comments:




 
 


 

see you ...