# 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  where another notion of Chaos is proved for a.positive measure of parameters.  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.