I'd like to learn about the period-doubling route to chaos of the logistic family $f\_\lambda(x)= \lambda x (1-x)$ and got interested in the fine properties of the bifurcation diagram of this family as we vary $\lambda$.


On wikipedia <https://en.wikipedia.org/wiki/Logistic_map> they claim that for most parameters $\lambda \in [3.56995, 4]$ that $f\_\lambda$ is chaotic, except for some islands of stability.



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> Can we classify those islands of stability?
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Also it is claimed that if we zoom in around $\lambda=3.82843$ (the end of what they call the Pomeau–Manneville scenario) then we roughly recover the original bifurcation diagram.



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> Do you know a reference for the proof of this self-similarity?
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In case you are aware of any paper discussing the bifurcation diagram for the logistic map in detail, please let me know.



Fine structure of bifurcation diagram of logistic family
The questions you are asking are fundamental to the theory of one-dimensional dynamical systems. I would suggest starting with an introductory textbook, such as *An Introduction to Chaotic Dynamical Systems* by Devaney. Books with more in-depth results include *Iterated Maps of the Interval as Dynamical Systems* by Collet and Eckmann, and *One-Dimensional Dynamics* by Melo and van Strien.


By the way, the islands of stability you mention are related to [Sharkovskii's theorem](https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem) and [Milnor-Thurston kneading theory](https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory), both of which are covered in Devaney's book. The self-similarity is a result of something called "renormalization" (which as far as I know is not related to the concept with the same name in quantum field theory).


