(b) The point 1 - 1/s is fixed, and even though it is
unstable, starting exactly there the iteration stays there. To find other points not
converging to the stable 2-cycle, reverse graphical iteration from the fixed point.
This produces two infinite sequences of points (one accumulating at x = 0, the other at
x = 1) going to the fixed point, hence not converging to the 2-cycle. Of course, x = 0
and x = 1 behave as they did in (a), and so don't converge to the 2-cycle. The inverse
graphical iteration process is illustrated by the dashed lines in the figure below
(s = 3.4).
|