On this this wikipedia page, it is claimed that the iterative sequence $x_{n+1}=rx_n(1-x_n)$ (the logistic map) starting at a point $[0,1]$ and where $r$ ranges in $[0,4]$ behaves differently according to $r$. For example:

- For $r\in[0,1)$, $x_n \to 0$, for all $x_0$;
- For $r\in [1,2)$, $x_n \to \frac{1-r}{r}$, for all $x_0$;
- For $F<r<4$ (where $F$ is the Feigenbaum constant), almost all of the values of $r$ produce a chaotic behaviour.

As I tried for several hours to find a book where a full characterization (and proof!) of the different behviours of this iterative sequence, I did not find any proof, except for the easiest cases (namely $r \leq 2$, $r=4$ and somes cases with two or four limit points).

Can someone provide me with a reference where this iterative map is studied in its full extent?