Zone 2:
R > 1
| r1 | or | r2 | smaller than 1.





The behaviour of the system in this zone is no more "absolute" but can only be interpreted in terms of probability.

The graph below shows the evolution of the system for r1=1.25 and r2=0.81:

Log(|X(n)|) vs n



What is plotted is the logarithm of the nth iterate versus n.

We can see that the system still diverges to infinity but the main difference with the previous case is that the growth of X(n) is no more monotonic: as the ratio r2 is smaller than 1, the system does not always move away from the origin.

Actually, when the center B1 is choosen, the system effectively tends to diverge but at the opposite, when the center B2 is choosen, the iterates tends to get closer to the origin.

The final system is the result of the competition between theses two effects.

It is important to notice that the probabilities p1 and p2 play a large role in the behaviour of the system in this domain.
Actually, the ratio r2 can be very small (and hence B2 be very attractive) but if the probability of the associated function is not large enough, the system will still diverge.
 

What happends is that after a great number N of iterations, the first function is chosen N*p1 times and the second function is chosen N*p2 times.
Moreover, if | X(n) | is great compared to D=(B2-B1), we can roughly say that | X(n+1) | ~ | r | * | X(n) | with r=r1 or r2.
Thus, after N iterations and in average, we have | X(N) | ~ | r1 |Np1 * | r2 |Np2 * | X(0) |.
Consequently, if R = | r1 |p1 * | r2 |p2 is greater than one the system diverges with a probability equal to one.
At the opposite, if R is smaller than one the system diverges with a probability equal to zero (see next page).
 
 

Conclusion:
This case corresponds to R>1, and so, even if the system does not diverge in an absolute way, the probabilty to diverge is equal to one and we can say that there is still no attractor.
 



I'll miss you...