In this section we will consider IFS based on two functions but that are restricted to the real axe (functions are only homothetias and there are no rotations).
We can already say that in such a system, the behaviour of the system will be the result of the "fight" between the two convergence properties of each function: we can guess that when | r1 | and | r2 | will be greater than one, the system will certainly diverge and that we may find an attractor when the modulus of r1 and r2 will be smaller than 1.
But let's be more precise and give first results.
The formula for this system is:
First of all, we must choose the homothetias' centers B1 and B2. In fact, this parameter is not very important because every set (B1,B2) gives the same results that the reference set (-1;+1), modulo an affine transformation.
Thus, we will work on this reference set:
Then we must assign a probability p1 and p2
to each function.
The behaviour of the system is quite complex and we will explain the role of the probability further. Nevertheles, for most of the experiments that wil be presented, I chose to work with:
All we have to do now is to take different values for
r1 and r2 and to see what happens...
The picture above represents the domains where the IFS shows very different behaviour according to the value of the parameters (r1,r2).
At this point of the discussion, it is important to notice that the behaviour of the system is mainly governed by the number:
R = | r1 |p1 * | r2 |p2.
We can then remark that this graph is mainly divided into
two parts, separated by two hyperbolas of equations R=1.
In fact, when R>1 (outside the hyperbolas), the system diverges with a probability equal to one.
At the contrary, when R<1 (inside the hyperbolas), the system diverges with a probability equal to zero.
We can also refine our description by looking at | r1 | and | r2 | : if they are both greater than one, the system diverges absolutely (independently of p1 and p2), but if they are both smaller than one, the system has an attractor.
Finally this gives us 4 major situations to discuss:
* Zones 1 (Red) : | X(n) | becomes larger and larger in a monotonic way when n increases: there is no attractor.
* Zones 2 (Green) : | X(n) | still becomes larger and larger when n increases but the growth is no more monotonic. Any way, the probability of staying at a finite distance from a given point is equal to zero: consequently, there is still no attractor.
* Zones 3 (Purple) : X(n) travel on an infinite part of the real axe but reaches the infinity with a probability equal to zero. These are "pathological" (and actually not very interesting) cases where the existence of the attractor is not easy to decide.
* Zones 4 (Blue) : This is the most interesting part of the graph: the trajctory of the iterates X(n) is included in a compact part of the real axe and we will see that we can always define an attractor.
Of course, there are also "limit" situations when r1 or r2 crosses one of the boundaries separating the different zone mentionned above (by example when R=1, the system is equivalent to a brownian motion...).
We can also make several remarks:
* This graph seems to have two symetries defined by (r1,r2) -> (-r1,r2) and (r1,r2) -> (r1,-r2).What comes next is the precise analyse of what happens in each zone ...
This is only half the truth: the existence of the attractor is effectively independant of the sign of r1 and r2. However, when the attractor exists, its shape depends of the sign on r1 and r2...
* When we have r1=1 (or r2=1), the associated function is simply X(n+1)=X(n). When this function is chosen, the system is invariant, and we can say that the global system is equivalent to a single function system and we can apply the conclusion of the previous chapter.