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Many introductions to chaos start with logistic map $$ x_{n+1}=\lambda x_n(1-x_n) $$ and claim it is chaotic at some values of $\lambda$. Unfortunately, all proofs of chaos I saw were numerical and not rigorous. How does one prove that such a map is chaotic at a particular point in general (I suspect it may be well known in the field, but I couldn't find an anwer easily)?

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  • $\begingroup$ Which definition of "chaotic" are you using? Is an orbit dense in an interval chaotic enough? Or the orbit from almost all starting points dense? $\endgroup$ Mar 29, 2021 at 2:18
  • $\begingroup$ yes, orbit dense in an interval would be good. Or positive Lyapunov exponent. Or strangeness of the attractor. I am interested in the question, how one proves chaos mathematically. $\endgroup$
    – Pavlo. B.
    Mar 29, 2021 at 2:31
  • $\begingroup$ It can be proved for $\lambda=4$. For other values of $\lambda$, I'm not sure anyone has a rigorous proof, there may only be compelling numerical evidence. $\endgroup$ Mar 29, 2021 at 2:34
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    $\begingroup$ Renormalization and the computer assisted proof of Oscar Lanford are rigorous. Here is one beautiful paper $\endgroup$
    – Mittens
    Mar 29, 2021 at 15:02
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    $\begingroup$ Yes, that is my understanding too. Another direction is the work of [1] where another notion of Chaos is proved for a.positive measure of parameters. [1] Jakobson, M.V., 1981. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Communications in Mathematical Physics, 81(1), pp.39-88. $\endgroup$ Apr 1, 2021 at 16:07

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Here is the proof of typical dense orbits for $\lambda=4$:

Start with the doubling map $\quad \theta_{n+1}=2\theta_n \mod 1$ on $[0,1) \,$. By considering the binary expansion of $\theta_0$, the law of large numbers implies that almost all orbits of the doubling map are dense. Write $x_n=\sin^2(\pi \theta_n)$ and observe that it satisfies the given recursion with $\lambda=4$. It follows that this map has typically dense orbits in $[0,1]$ as well.

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