24. (a) Looking at the Tent Map bifurcation diagram for s > 1, we see the top of
the diagram has coordinate s/2. The bottom of the diagram is the first iterate of this,
hence s - s2/2. From the diagram we see this is less than 1/2. The
bottom of the upper branch (before band-merging) of the diagram is the iterate of
this, hence s*(s - s2/2) = s^{2} - s^{3}/2.
Finally, the top of the bottom branch (before band-merging) is the iterate of this,
hence s - s*(s^{2} - s^{3}/2) =
s - s^{3} + s^{4}/2. The band-merging occurs where these
last two quantities are equal:
s^{2} - s^{3}/2 =
s - s^{3} + s^{4}/2. That is,

0 = (s/2)*(2 - 2*s - s^{2} + s^{3}).

(b) It is easy to see s = sqrt(2) satisfies this equation:

2 - 2*sqrt(2) - (sqrt(2))^{2} + (sqrt(2))^{3}
= 2 - 2*sqrt(2) - 2 + 2*sqrt(2) = 0.

Return to Exercises