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.