When r_{1} and r_{2} are both positive,
this case is quite simple : as soon as X(n) leaves the segment [B_{1};B_{2}],
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 p_{1} and p_{2}) 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.
b_{1}=(B_{2}-B_{1})*( | r_{2}
| +1)/( | r_{1} | * | r_{2} | -1)
and
b_{2}=(B_{2}-B_{1})*( | r_{1}
| +1)/( | r_{1} | * | r_{2} | -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 r_{1}=r_{2}=-1.1. For these values, with B_{1}=-1 and B_{2}=+1, we have b_{1}=b_{2}=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 B_{1} and B_{2}.
However, as soon as | X(n) | becomes greater than 21,
the system still oscillates (because r_{1} and r_{2} are
negative) but we clearly have a monotonic increase of | X(n) |.
b_{1}=(B_{2}-B_{1}) * | r_{1} |.
We can remark that this case is less "stable" than when both r_{1} and r_{2} are negative and hence, there is of course no attractor.
The graph below shows what happends when r_{1}=-1.2
and r_{2}=1.05 (we have then b_{1}=2.4):
A soon as the red curves leaves the interval [-3.4;1]
the modulus of X(n) increases in a continuous way.
Conclusion:
We have clearly seen that in all these case ( | r_{1}
| > 1 and | r_{2} | > 1 ), there can never be an attractor as the
system diverges to inifinity in a monotonic way.