The Graph of F2


Note that for the graph of F, when the attracting fixed point p is such that F'(p) < 0 then there is a partner to p, call it p1, such that F(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 . Observe that in this region, F2 looks like an upside-down version of F.

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 RF has many of the same features as F:

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 will undergo the same period doubling bifurcation that F 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 = 3.83 there are period 3 points.


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