Proven chaos in logistic maps

Question

For a logistic map using $f_r(x)=rx(1-x)$, what values of $r$ and starting $x$ are guaranteed (i.e., with an accepted proof) to be chaotic? I mean "chaotic" in the loosest sense: The limiting sequence of $x$ does not tend to a finite number of points.

I am currently using $r=3.8$ and starting $x=0.501234567890123456789$, but have only tested through 10,000 iterations. What is the probability that I am chaotic?

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EDIT: Below are new results (with 4,000,000 bits of precision to avoid any rounding problems) for 2,000,000 iterations (showing matches to "0.72224", the end's most significant digits). So, I believe it is fair to say that there are 3 possible cases:

1. There is no limit cycle (through infinity),

2. There is a limit cycle of at least 1,105,578 points, or

3. There is a smaller limit cycle but any "two points chosen from the first 2,000,000 points" are not both within one limit point's attractor zone.

#2 seems the most unlikely. #3 seems unlikely simply because I chose such round numbers from the start. According to answers here, however, it does seem like the probability for #1 is not 100%. Maybe someone can put my statements here into proper mathematical language and clarify this better.

      n: x_n
-----------------------
  53951: 0.7222489331
  66539: 0.7222408270
  68976: 0.7222441979
  75138: 0.7222495664
 120428: 0.7222473699
 134963: 0.7222441673
 235912: 0.7222411119
 395643: 0.7222459509 closest greater value
 417062: 0.7222404139
 462528: 0.7222468852
 472142: 0.7222408308
 645137: 0.7222474275
 679584: 0.7222492244
 731458: 0.7222410420
 761284: 0.7222468048
 891274: 0.7222442328
 894423: 0.7222448046 closest lower value
 935412: 0.7222498698
1110025: 0.7222446506
1220483: 0.7222447341
1222255: 0.7222485044
1269796: 0.7222407187
1301786: 0.7222439936
1422147: 0.7222488714
1431959: 0.7222457998
1503338: 0.7222445272
1509878: 0.7222404127
1568206: 0.7222447453
1569439: 0.7222415020
1612039: 0.7222497768
1634269: 0.7222406207
1642044: 0.7222450907
1791569: 0.7222487370
1865739: 0.7222420900
1879844: 0.7222427753
1902889: 0.7222493257
2000000: 0.7222453893 end

Answer 1

By choosing $r=4$, the logistic map is topologically semi-conjugate to the doubling map on $\mathbb{R} / \mathbb{Z}$: the solution takes the form $ x_{n}=\sin ^{2}(2^{n}\theta \pi)$. Thus any irrational $\theta$ gives a chaotic orbit.

Answer 2

For all values of $r > r_0 \simeq 3.5699...$, the topological entropy of the logistic map is strictly positive [1]. That means that there is an uncountable set of points whose orbit accumulates on a compact set that is not countable, ie there is an uncountable set of points which are chaotic in your sense.

Note that the term "chaotic" is often misused. When it comes to the logistic family, the term "chaotic parameter" is often used with a different meaning, namely the existence of an non-uniformly hyperbolic SRB measure attracting a set of points of full Lebesgue measure. It is known that the set of such parameters has density going to one as the parameter goes to $4$, but it is also known that its complement is open and dense in the interval $[0,4]$.

It is in general pretty hard to determine if a given point for a given parameter has an orbit converging to a periodic orbit. On the other hand, it is usually feasible to exhibit a parameter in a given interval and an initial value in a given interval whose orbit has a specific behavior.

For your particular case, I would be surprised if an answer can be given because for the parameter 3.8 there are infinitely many periodic points and your initial value may fall on such a point after a while, but due to rounding errors, this would not be caught by the computer. Or instead a non periodic orbit may end up in a loop due to such errors.

[1] the graph of the entropy as a function of the parameter appears for example in the 2015 article of Bruin and Van Strien https://www.ams.org/journals/jams/2015-28-01/S0894-0347-2014-00795-5/ Monotonicity of the entropy goes back to work of Douady (1995) and Milnor, Thurston (1988).

Answer 3

There is a nice article by Mikhail Lyubich in the October 2000 edition of Notices of the AMS, "The Quadratic Family as a Qualitatively Solvable Model of Chaos" that provides a nice summary of rigorous results regarding logistic maps. The main results relevant to your question are the following (some of these are in Lyubich's paper, others are not).

1. Let $S$ denote the set of parameter values $r$ for which the map is "stochastic" in the sense that there is an invariant probability measure that is an absolutely continuous measure with respect to Lebesgue. Then the Lebesgue measure of $S$ is positive (so a positive proportion of parameter values lead to stochastic behavior); this was first proved by Jakobson (Comm. Math. Phys. 1981), see also work by Collet and Eckmann (Comm. Math. Phys. 1980, ETDS 1983) and by Benedicks and Carleson (Annals 1985). For $r\in S$, Lebesgue-a.e. $x$ has an orbit that is distributed according to this measure, and in particular its $\omega$-limit set is a union of intervals.

2. Let $R$ denote the set of parameter values $r$ for which the map is "regular" in the sense that there is an attracting periodic orbit. Then $R$ is open and dense in parameter space, and $R\cup S$ has full Lebesgue measure: Lebesgue-almost every $r$ gives a map that is either regular or stochastic in the above senses.

3. Regarding specific values of $r$, say that the parameter $r$ is a Misiurewicz point if the critical point for the map is pre-periodic but not periodic. Such parameter values are stochastic in the sense above (Misiurewicz 1981, Pub Math IHES). Note that in particular $r=4$ is a Misiurewicz point. If I recall correctly, the proof that $S$ has positive Lebesgue measure involves a process called "parameter exclusion" near Misiurewicz points and actually shows that each Misiurewicz point is a Lebesgue density point for $S$.

4. It is an interesting question to get rigorous bounds on the Lebesgue measure of $S$. I don't know the full story here, but there is a paper by Tucker and Wilczak (Physica D 2009) and a more recent one by Golmakani, Koudjinan, Luzzatto, Pilarczyk (Chaos 2020) with some results.