# The Graph of F2

 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. , where 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 3 points. Next: Follow the "Zoom In" link.