When r1 and r2 are both positive,
this case is quite simple : as soon as X(n) leaves the segment [B1;B2],
we have | X(n+1) | > | X(n) |.
Moreover, it is sure (meaning with a probability equal to one) that if we wait long enough, one iterate X(n) will leave this interval.
The graph below shows this phenomenon.
Consequently, the system diverges absolutely (meaning independently of the probabilities p1 and p2) to infinity.
There is no attractor.
As it was said previously, the behaviour of the system
is quite similar in the other "symetrical" domains.
However, few differences can be noticed.
This is why I thought it was useful to discuss these cases, just to convince ourselves that these differences were very small.
b1=(B2-B1)*( | r2
| +1)/( | r1 | * | r2 | -1)
b2=(B2-B1)*( | r1 | +1)/( | r1 | * | r2 | -1).
Moreover, as in the first case, we are sure that if we
wait long enough, X(n) will leave this segment and so, we can conclude
that for any initial condition X(0), the system diverges and that there
is no attractor.
The pictures below show an example for r1=r2=-1.1. For these values, with B1=-1 and B2=+1, we have b1=b2=20.
The first one clearly shows that after a finite number
of iteration (~150), the growth of | X(n) | becomes monotonic.
We can see on the second graph what happends during the
first iterations. The interaction between the two functions makes X(n)
oscillating in the neighbourhood of B1 and B2.
However, as soon as | X(n) | becomes greater than 21, the system still oscillates (because r1 and r2 are negative) but we clearly have a monotonic increase of | X(n) |.
b1=(B2-B1) * | r1 |.
We can remark that this case is less "stable" than when both r1 and r2 are negative and hence, there is of course no attractor.
The graph below shows what happends when r1=-1.2
and r2=1.05 (we have then b1=2.4):
A soon as the red curves leaves the interval [-3.4;1] the modulus of X(n) increases in a continuous way.
We have clearly seen that in all these case ( | r1
| > 1 and | r2 | > 1 ), there can never be an attractor as the
system diverges to inifinity in a monotonic way.