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A slightly more theoretical question for you all. Recently I was looking at the logistic map and the resulting bifurcation diagram (shown). Wikipedia says that prior to roughly r = 3.56995, there is a period-doubling cascade that goes from 2 to 4 to 8 to 16, etc. I assume this continues on for infinity. My question is, is there a concrete way to prove this? Ie., prove that there is an infinite amount of even periods prior to roughly r = 3.56995. I was planning on using an induction proof, start proving that the logistic map stabilizes for values 1 < r < 3 (I can attach the completed proof for this case if needed). However, I haven't quite been able to extend this enough to make it complete in induction. Any ideas?

Alex Mac
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    I have never studied the proof of this period-doubling, but I would be surprised , if induction would already be sufficient to prove this. – Peter Aug 22 '21 at 15:09
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    It can be proved, for sure. I haven't studied the proof, but I doubt it's short and simple enough to be given in an answer here (but I'll be happy if someone proves me wrong about that). You can find plausibility arguments, at least, in Devaney or Strogatz. – Hans Lundmark Aug 22 '21 at 16:09
  • Hi, thanks for the responses! Can you give a brief outline as to what or how these proofs approach this problem? I don't exactly want to buy a text book for it if I can avoid it (though lets be real, we've all spent too much money on math text books already) – Alex Mac Aug 22 '21 at 21:38
  • The only place that I have seen this proven is in Feigenbaum's proof on the universality of the Feigenbaum constant. http://chaosbook.org/extras/mjf/LA-6816-PR.pdf The proof of the existence of the Feigenbaum constant implies that there are infinitely many period doubling bifurcations. – mwalth May 09 '22 at 18:51

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