How can one prove chaos in logistic map?

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)?

• 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? Mar 29 at 2:18
• 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. Mar 29 at 2:31
• 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. Mar 29 at 2:34
• Renormalization and the computer assisted proof of Oscar Lanford are rigorous. Here is one beautiful paper Mar 29 at 15:02
• 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. Apr 1 at 16:07

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.