Note that for the graph of F_{µ}, when the attracting fixed point p is
such that F^{'}(p) < 0 then there is a partner to p, call it p_{1},
such that F_{µ}(p_{1}) = p and p_{1} < p. Note further
that p_{1} = 1  p must hold.
Look at the graph of F^{2} in the domain [p_{1}, p], for a given value of µ. Observe that in this region, F^{2} looks like an upsidedown version of F_{µ}. This can be made precise with the renormalization operator, R. , where
Note that L takes p to 0 and p_{1} to 1. Further note that the renormalized function RF_{µ} has many of the same features as F_{µ}:
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 RF_{µ} will undergo the same period doubling bifurcation that F_{µ} 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
3 points.
