Note that for the graph of Fµ, when the attracting fixed point p is
such that F'(p) < 0 then there is a partner to p, call it p1,
such that Fµ(p1) = p and p1 < p. Note further
that p1 = 1 - p must hold.
Look at the graph of F2 in the domain [p1, p], for a given value of µ. Observe that in this region, F2 looks like an upside-down version of Fµ.
This can be made precise with the renormalization operator, R.
Note that L takes p to 0 and p1 to 1. Further note that the renormalized function RFµ has many of the same features as Fµ:
The proofs of the above statements are straightforward.
It is clear that after renormalizing we have a function just like we started with, and this new function RFµ will undergo the same period doubling bifurcation that Fµ underwent. Thus, we have a period doubling route to chaos.
Do observe that there are also saddle node bifurcations in there, which produce
odd periods. For example, in the neighborhood of µ = 3.83 there are period