Talk:Logistic map

New Content and content from translation

Part of the content of this article was translated from the japanese version with extra editing Template:Translated Unfortunately a lot of citations from the japanese article are not reusable (e.g. they refer either to japanese university teachers, japanese books or japanese translations) and the international ones needs to be individually checked.

The article only assumes some knowledge of high school calculus. It is of high importance given it's is a prototype system, and it can be considered a gentle introduction to chaos and dynamical systems to trigger interest for people in a high school. Therefore a very slow paced style is used, with a lot of visual diagrams for understanding, in similar style to the japanese article. In this sense the article can also be considered a starting point for a high school project.

There are some "inspirational" references, to turbulence, phase transitions and other systems that have similar phenomena, that are left on purpose without the relevant mathematics. The idea would be also to give a divulgation perspective on these subjects such as the relevance of universality and renormalization both in turbulence, and phase transitions. This is a bit of a recent (2020s) trend in education and research to see fluid dynamics analogies including solitons, chaos and turbulence as foundational to many-particle phenomena and strongly correlated systems that are little understood in condensed matter physics e.g but also in Quantum Field Theory and ultimately also quark gluon plasma, which exibits some shape of "poorly" understood phase transitions. The article, as usual, shall not include original research but is important to show also to young students what are the open problems and recent trends in physics.

Further editing needs to be kept consistent with these principles. Flyredeagle 03:46 Dec 19, 2024 (UTC)

Very old discussions

The logistic map is related to the Mandelbrot set by the equation c=(1-(r-1)²)/4.

I can't follow the algebra here. Mandelbrot is z[n+1] = z[n]^2 + c and logistic is x[n+1] = r x[n] (1-x[n]). I don't think the formula given above converts these formulas into each other. AxelBoldt 02:24 Sep 30, 2002 (UTC)

I agree.

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Rewrite the logistic map as x_n+1 = r ( 1/4 - (x_n-1/2)²) and use the change of variables: z_n = r(1/2 - x_n). Result: z_n+1 = z_n² + c with c=(1-(r-1)²)/4. Abelian (talk) 21:55, 12 August 2009 (UTC)

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Are there any other formulae that have this property.

I.e., you have the Mandelbrot set and the logistic map being a degree 2 polynomial. Are there any interesting examples with degree 3? Degree 4? Other types of functions?

Yes. If you iterate practically any function that's not linear (not a straight line), then there will be some starting values that give chaotic (although, of course, perfectly deterministic) results. The FractInt software comes with lots of them. -- DavidCary

(Should this answer go on the article page ?)

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There seems to be a problem with this equation:

    x_{n+1} = r x_n (1 - x_n), 

One of the quantities X sub n on the right side of the equation should be an X sub 0. I'm not sure which one, though.

Nope. Both on the right are supposed to be x_n. -- DavidCary

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Regarding "also given access to perfect computation...". This statement is logically false, because a logistic map, in principle, operates on un-computable numbers. This means that even a theoretically ("in principle") "perfect" computer could not compute the vast majority of trajectories. And by vast majority, I mean 99.999999999...%. For a formal definition of computability, see Turing's essay "On Computable Numbers".

-- Kevin Baas 20:11, 26 Feb 2004 (UTC)

I was going to fix this paragraph myself, and I had much trouble, because when ever I encountered a "not the same as..." I thought for a minute and eventually answered "Yes it is." The proper way to speak of predictability/unpredictability is in information-theoretic terms. In information theoretic terms, one speaks of probability, noise, and divergence. A stochastic system is probabilistic, but may still very well be a continuous stochastic system, that is, have an information dimension of one, and ultimately represent the same amount of information as a chaotic system. Indeed, one can have a stochastic chaotic system, with an arbitrary rate infromation decay/divergence. Chaotic and stochastic are orthogonal classifications. A linear(non-chaotic) and non-stochastic system simply evades the question of information and therefore predictability altogether. We must therefore disclude it from any concept of predictability or randomness. Indeed, randomness already discludes all non-stochastic systems. Where a system is "random", it has "divergence", times a scalar. A system may have divergence distributed evenly in resepct to a real-valued parameter. In this case, one simply adds a "noise" term: + N. This is so-called "randomness". In a chaotic system, on the other hand, the divergence is concentrated by a point. Regardless, one has convergence and divergence, and in either case noise is noise is noise. i.e. randomness is divergence is noise. In terms of "causuality" in the markov sense, there is really no fundamental discriminatory factor. It still remains possible, ofcourse, for one to make a (superficial) mathematical distinction, but one should not look for any ontological distinction - it is ontologically unnecessary for there to be a distinction.

But I have not yet been complete enough, one might still raise the objection that a chaotic system is fully specified by a finite set of symbols and is non-stochastic. However, my point is that this is completely irrelevant. The symbols are arbitrary.

A chaotic is neccessarily a non-equilibrium system; chaotic systems only exist where there is a flow of energy, and thus also neccessarily, a flow of information. The "unpredictability" of a chaotic system is part of this flow, just as the "noise" in a stochastic system is an influx of information. The difference is simply whether the distribution of this influx - this flow - is flat with respect to a given measurement.

But there is also the philosophical problem of "noise" as extrinsic "unknown" (determinism) - the scientific deterministic assumption - vs. noise as an in-itself (free-will). But in either case there is, inextricably, noise, and thus noise, since it cannot be reduced, replaced, or removed, must ultimatly be in-itself, regardless of whether it is "extrinsic" or "intrinsic". (The question was never what name we give to the difference of the being-of-noise, but rather the simply differential character of the difference itself.) The more fundamental question is not intrinisic vs. extrinsic, but in-itself (essential) vs. phenomenological. But the two positions are informationally equaivalent/indistinguishable: one nonetheless has "noise", "information", and everything else. There is no way to statistically distingush among the consequences beyond a mere quantification of the "noise rate". -- Kevin Baas 06:21, 28 Feb 2004 (UTC)

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"With r between 3 and 1+√6 (approximately 3.45)..." It seems like there's a typo in the 1+6, but I'm not sure what it should be --Kevinatilusa

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Why does it keep saying independent of inital seed? It seems like (r-1)/r is a solution no matter what r is (r>=1). For example if r was 4, you wouldn't get much chaotic behavior if you started with .75 .Is the talk about period and chaotic behavior talking about a general seed such as a transcendental one? Hiiiiiiiiiiiiiiiiiiiii 00:41, 23 May 2006 (UTC)

note

if r = 4, the logistic map appears to be solvable: set

$\varphi ={\cos }^{-1}(1-2x_0)$

then if

$x_{n+1}=4x_n(1-x_n)$

we have

$x_n=\frac{1}{2}-\frac{1}{2}\cos (2^n\varphi )$ which can be easily checked. Scythe33 20:55, 18 December 2006 (UTC)

Yes, that's correct. There are also explicit solutions for r = 2 and r = -2 More information can be found here: [1] Scribblesinmindscapes 16:29, 10 January 2007 (UTC)

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Chaos (or period 3) begins at precisely r = 1 + Sqrt(8)

There are methods for proving that period 3 begins at exactly r = 1 + Sqrt(8). I can provide a published proof if necessary, but also did the proof myself as a presentation in a chaos theory class. I think this section could benefit by mentioning this. 68.55.58.255 07:41, 23 May 2007 (UTC) Please give a reference to published proof in article and if I'm not wrong good place for a your proof is wikibooks ( with link to wikibooks in article). --Adam majewski (talk) 14:37, 13 August 2009 (UTC)

Biology

It would be interesting to read about connections to biology. Are there interesting observations in nature predicted by this? (Most interesting seems the range 3-3.45) —Preceding unsigned comment added by 130.242.107.211 (talk) 02:55, 7 April 2008 (UTC)

Logistic map with WEBPLOT

There is another way, using WEBPLOT program:

XMIN : 0
XMAX : 1
XSCL : .1
YMIN : 0
YMAX : 1
YSCL : .1
XINIT : $A2/.5+
WEBSTART : 0
WEBSTOP : 50
CURRENT : 1'-'*$B1+2**
REFERENCE : '

--zzo38(✉) 01:24, 23 April 2010 (UTC)

Higher Resolution Orbit Map

I created a higher resolution orbit diagram for the logistic map and was wondering if indeed it is worth putting here.

File:Logistic Bifurcation map High Resolution.png

High Resolution Orbit Diagram

File:LogisticMap BifurcationDiagram.png

Orbit Diagram in Article

Please let me know what you think. Thanks! Efecretion (talk) 08:39, 12 September 2011 (UTC)

could you put the code ( in commons) and describe algorithm ? --Adam majewski (talk) 19:12, 14 September 2011 (UTC)

I could upload the Mathematica .nb file. The algorithm is similar to the one described in the old image. Where in commons would I put the info? --Efecretion (talk) 06:28, 15 September 2011 (UTC)

File:LogisticMap EarlyIterates.png

Early iterates

I scribbled over Image:LogisticMap BifurcationDiagram.png tracing the first few iterates in order to give a visual explanation of the behavior of the chaotic "shadows". Not sure where/if it's worth publishing it... so I just mention it here, for now. ale (talk) 18:02, 10 October 2018 (UTC)

Inconsistent diagram

The first graph on the page (the animation) has mislabeled axes. The horizontal axis should be "n" to be consistent with the equation given in the article. Took me forever to figure out what it was talking about. I'd do it myself but I don't know how to make animated gifs.

205.175.97.169 (talk) 06:56, 19 May 2014 (UTC)

GIF, axes

The right limit for the t axis is too big. I suggest making it (0,75) instead of (0,200). It's really too difficult to see the transient dynamics in the current GIF. MATThematical (talk) 22:42, 22 May 2014 (UTC)

How about this?

File:Logistic map animation.gif

Regards, Snaily (talk) 08:27, 9 July 2014 (UTC)

Much better. I love it! Can you add it to the main page MATThematical (talk) 15:42, 8 September 2014 (UTC)

Thanks Matt, glad you like it. I scaled it down a bit and put it in. - Regards Snaily (talk) 05:15, 15 September 2014 (UTC)

r values below 0

Should there be description on what happens if the value of r is below 0? --Artman40 (talk) 13:58, 20 October 2014 (UTC)

The population goes negative, which doesn't really make sense as a population model? I imagine that's why nobody seems to have bothered to consider this case. Double sharp (talk) 10:09, 27 March 2015 (UTC)

I was wrong about this: MathWorld has a page considering r = −2, which also admits an exact solution. Double sharp (talk) 12:09, 31 August 2018 (UTC)

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Badly labelled graphs

File:Logistic map animation.gif

These two graphs have to be some of the worst I've ever seen. They are so badly labelled, and give no idea to a reasonable user as to what the they are showing (the right graph specifically). "Parameter" should at least be labelled "parameter, r", and none of t, T, f and k are anywhere defined properly. And "[n] t=nT" is not how you properly label an axis. First, either it is "t" or it is "n" (and not "[n]" either), and secondly, some description would be nice. I've seen better graphs than this on vixra. It needs to be sorted out or removed completely, unfortunately. Krea (talk) 20:58, 8 September 2019 (UTC)

Missing text

Several parts of this article are evidently missing text. For example under "Research history" there is Template:Xt followed by an equation, after which is an entirely new sentence. This is gobbledegook; surely something has got lost. Hairy Dude (talk) 03:05, 19 May 2025 (UTC)