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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.

Can we classify those islands of stability?

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.

Do you know a reference for the proof of this self-similarity?

In case you are aware of any paper discussing the bifurcation diagram for the logistic map in detail, please let me know.

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  • $\begingroup$ Are there no papers cited at that Wikipedia page? $\endgroup$
    – Gerry Myerson
    Commented Dec 9, 2019 at 11:24
  • $\begingroup$ @GerryMyerson There is a paper listed for a particular value ($\lambda = 1+\sqrt{8}$) where an island of stability starts. However, none for where they are located in general. No paper is mentioned for the self-similarity. $\endgroup$
    – Severin Schraven
    Commented Dec 9, 2019 at 13:25
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    $\begingroup$ mathoverflow.net/q/361637/37099 $\endgroup$
    – Adam
    Commented May 20, 2022 at 9:59

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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 and Milnor-Thurston 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).

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  • $\begingroup$ Indeed, I am reading Devaney's book and there he makes the claim that those islands exist without giving a proof. I do have Collet and Eckmann's book on the shelf in my office, will check it out tomorrow. $\endgroup$
    – Severin Schraven
    Commented Dec 9, 2019 at 21:36
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    $\begingroup$ I must admit that I do not see the connection between the islands of stability and the kneading theory. Maybe I need to have another look at the relevant chapter, but it seemed to me that Devaney only elaborates on the period-doubling cascade and not on the connection to those islands. $\endgroup$
    – Severin Schraven
    Commented Dec 9, 2019 at 21:40

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