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?

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    $\begingroup$ Saber N. Elaydi's textbook, Discrete Chaos, goes into some detail about this map. Whether you'll consider it to be the "full extent", I don't know. $\endgroup$ May 29, 2020 at 10:44
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    $\begingroup$ @GerryMyerson thank you. In fact I already encountred books where some "parts" of the proof are given, but never the difficult parts (intervals where chaos occurs, or when chaos disappear). It is really strange that there no theorem of the form "for $r \in I_1$ then [...] occurs, for $r \in I_2$ then [...] occurs, etc.". $\endgroup$
    – J.Mayol
    May 29, 2020 at 19:44
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    $\begingroup$ It's possible that for some of what you want there are no theorems, just numerical evidence. $\endgroup$ May 29, 2020 at 21:59
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    $\begingroup$ Try the book "Iterated Maps on the Interval as Dynamical Systems" by Collet and Eckmann. springer.com/gp/book/9780817649265 $\endgroup$ Jun 10, 2020 at 19:55

2 Answers 2


If you change the form of your map to $z_{n+1} = z_n^2 + c $ ( conjugation) and take only real values of c ( real slice of Mandelbrot set) then you will find the answers in papers by G. Pastor, M. Romera. Here is for example : Calculation of the Structure of a Shrub in the Mandelbrot Set

The part from 0 to Feigenbaum point in your map is a periodic region where period doubling cascade occurs. In c plane it is from 0.25 to F

The part from F to 4 in your map is a Mandelbrot set antenna. It's structure is described in that paper :

IMage from paper

Look also for:

Part of parameter plane with mini Mandelbrot sets for periods 1, 3, 9, 27, 81, 243. External rays are red.

"Many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler



Lyubich wrote a nice short survey on this: "The quadratic family as a qualitatively solvable model of chaos", in the October 2000 issue of Notices of the AMS. At just 11 pages long, that survey is fairly terse but it does hit the most important points and cites some of the original papers where things are proved, so you can follow the references to find the proofs. As a general rule the proofs are fairly involved and require one to first build up quite a bit of machinery.

There is also a longer survey by Graczyk and Świa̧tek from around the same time that has more details and a more comprehensive list of references:

Graczyk, Jacek; Świątek, Grzegorz, Smooth unimodal maps in the 1990s. (Survey), Ergodic Theory Dyn. Syst. 19, No. 2, 263-287 (1999). ZBL0941.37024.


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