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The logistic map typically converges to a "carrying capacity", or output value across iterations, at r values (the growth rate) under 3. After 3, the value the system converges to bifurcates multiple times until r reaches 3.57. After 3.57, it exhibits entirely chaotic behavior and I'm not sure why this exact value is so significant. I've seen some sources discuss bifurcation past 3 in other similar systems, but 3.57 seems so random to me.

To elaborate on the topic and my understanding, the bifurcation diagram is derived from the logistic map (seen on the wikipedia citation) by incrementally adjusting the growth rate 'r' and plotting the system's converging values (x sub n+1). Past r=3, the system bifurcates and oscillates between 2 values instead of one and continues to do this, going through periods 4, 8, 16...etc., until r=3.57 where it exhibits chaos and does not converge. From what I understand, the chaos and bifurcations originate from the alterations to the system's points of stabilization and equilibrium. Whatever property of non-linear dynamical systems or the logistic map that causes this is unknown to me as much of the generalizations go over my head. The only other significant point to add is that the bifurcation diagram has consecutive stable periods with a ratio equal to the feigenbaum constant ~4.6692 and the diagram is a partial projection of the Mandelbrot set on a 3rd axis.

Main sources: Contributors to Wikimedia projects. Logistic Map - Wikipedia. Wikimedia Foundation, Inc., 19 Aug. 2024, en.wikipedia.org/wiki/Logistic_map.

Logistic Map -- from Wolfram MathWorld. wolfram.com, 7 Nov. 2024, mathworld.wolfram.com/LogisticMap.html. Alonso-Sanz, R., et al.

“Bifurcation and Chaos in the Logistic Map with Memory.” Int. J. Bifurc. Chaos, Dec. 2017, semanticscholar.org/paper/….

Also Steven Strogatz book "non-linear dynamics and chaos" and Veritasium's video on the bifurcation diagram.

Juan Pablo
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    Please Use MathJax formatting – Tnol Nov 07 '24 at 00:51
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    Please give references, so that the explicit logistic framework is shown, (which is very useful for the part of the community that never saw this instance of chaotic system, or saw this occasionally, without a systematic approach,) and show your level of understanding, the place where $3.57$ comes into play, and so on... This site is designed not only as a quick one time Q&A-site, so that that one person asking gets a quick answer, and that's it, we can waste the question and the answer, instead, it is building knowledge, step by step, references inside MSE become important. – dan_fulea Nov 07 '24 at 01:13
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    Roughly, it's because the period doubling cascade stops at approximately 3.57! – Mark McClure Nov 07 '24 at 01:15
  • Okay, sorry for my mistake and misunderstanding. I will add references and more details soon so that I'm using the forum correctly. – Juan Pablo Nov 07 '24 at 01:58
  • I think the Alligood book Chaos: An Introduction to Dynamical Systems probably discusses this in detail. It's very readable. – MJD Nov 07 '24 at 03:23
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    It is not the case that the logistic map is chaotic for all $r>3.57$. There are "windows" of non-chaotic behavior beyond that value of $r$. – Gerry Myerson Nov 07 '24 at 03:27
  • @GerryMyerson The windows do, actually, exhibit chaos. If $r$ is chosen from the period 3 window, for example, then $f_r$ will certainly exhibit chaos, by a famous theorem. Within a window, that chaos is exhibited on a Cantor set and the critical point is attracted to a periodic orbit that lies outside the Cantor set. We can see the windows in the bifurcation diagram precisely because the technique of iterating from the critical point is searching for stability. It misses chaotic behavior that may, nonetheless be present. – Mark McClure Nov 07 '24 at 13:11
  • @Mark, I'm aware of the Li & Yorke "Period three implies chaos" paper. It was published before there was general agreement on what exactly "chaos" meant. According to the Devaney definition that was later generally accepted, period three does not imply chaos. If there is an asymptotically stable periodic point, there is no chaos. Chaos on a Cantor set doesn't qualify. – Gerry Myerson Nov 07 '24 at 13:26
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    @GerryMyerson That's funny - I learned this from Devaney. :) – Mark McClure Nov 07 '24 at 13:32
  • @GerryMyerson More specifically, section 11.3 of the first edition of Devaney's First Course in Chaotic Dynamical Systems describes very clearly how chaos arises on a Cantor set within the period 3 window. A similar analysis applies to the other periodic windows as well. I was simply trying to point out there is a quantitative distinction between the behavior for $r<3.56$ and $r$ within the periodic windows. – Mark McClure Nov 07 '24 at 14:19
  • You are, of course, correct that Period Three paper was published prior to Devaney's definition. That definition, though, is stated for metric spaces and certainly includes the possibility of a Cantor set. Your claim that Cantor sets don't qualify is quite curious given their centrality in real analysis and dynamical systems. – Mark McClure Nov 07 '24 at 14:20
  • @Mark, I take your points. There's nothing in the original question that indicates to me that OP is interested in chaos on Cantor sets (or even knows what a Cantor set is). Chaos in the windows is a very different beast from chaos at, say, $r=4$, and I took the latter to be what OP had in mind when using the term. If I have underestimated or misled OP, my apologies. – Gerry Myerson Nov 07 '24 at 21:20
  • Any thoughts, Juan, on the answer I posted? – Gerry Myerson Nov 08 '24 at 11:59
  • I was aware of the windows of “non-chaotic” behavior that has odd periods and do clarify I’m completely unaware of Cantor sets and what they are and even get lost in the idea of points being attractive so that’s a whole different issue. I saw the paper you mentioned as well about the period of 3 so that’s fine. – Juan Pablo Nov 08 '24 at 20:11
  • OK. So, Juan, does my answer clear things up for you? or is there something in your question that still needs to be addressed? You're not going to get the kind of answer you want if you don't let us know whether anything is missing in the answer you have received. – Gerry Myerson Nov 08 '24 at 21:03

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I'm not sure what more can be said beyond what you already know. I can add a few more decimals: the number you give as $r=3.57$ is given to $95$ decimals at https://oeis.org/A098587, along with some links to the literature that you may find interesting. The Feigenbaum constant, your $\delta=4.6692$, is given to over $1,000$ decimal places at https://oeis.org/A006890, again with links to the literature.

The $r$-constant is specific to the logistic map. It is the endpoint of the period-doubling cascade, and I don't think it has any independent meaning; it's just the limit of the values of $r$ where periods of lengths $2,4,8,16,\dotsc$ occur. For a different dynamical system that displays period-doubling, there will be a different limiting value of $r$, specific for that dynamical system.

The Feigenbaum constant, which is a certain limit based on those period-doubling appearances, is different: it's universal, meaning that any dynamical system that shares a few simple properties with the logistic map will lead to the same number $\delta$, even if it has a very different sequence of $r$-values.

In addition to the references at the oeis pages, you could see Elaydi, Discrete Chaos, and Brown, A Modern Introduction to Dynamical Systems.

Gerry Myerson
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  • It might be fun to include all 95 digits from OEIS in your answer: 3.5699456718709449018420051513864989367638369115148323781079755299213628875001367775263210342163 – Ben Bolker Nov 07 '24 at 13:28