The sequence is the solution of the difference equation...
(1)
... with some sort of 'initial value'
. We don't have information regarding r and, only for demonstrative purpose, we will suppose that
. The 'fixed point' of the sequence are the solution of the equation
that are...
(2)
With the hypotheses we have made
is a
repulsive fixed point [i.e. a point from which the
tend to diverge...] and
an
attractive fixed point [i.e. a point to which the
tend to converge...]. The figure is the case
so that is
...
From the figure we can derive the asintotic behavior of the
as function of
. In particular...
a) if
or
the sequence will diverge to
...
b) if
or
the sequence will converge to
c) if
the sequence will converge to
without oscillations...
Kind regards