Talk:Chaos theory/Archive 1

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Strange attractors

The Talk:Attractor page has a discussion on whether strange attractors need to be fractal. The difficulty resides with the definition of a strange attractor. The term strange attractor is often used as synonym for chaotic attractor with the assumption that fractional-dimensional attractor is the same as chaotic attractor. They are not. This topic was often discussed in the sci.nonlinear and is now a FAQ (question 2.12) for that newsgroup.

I have made a small edit in the main article to reflect this point. XaosBits 03:53, 9 May 2005 (UTC)

Chaos washing machines??

Does the section on Chaos Washing Machines really belong in this article? It sounds like a bunch of marketing hooey and self-promotion to me; I nominate that that whole section be moved to its own article if not outright delete. It sure doesn't sound encyclopeadic to me. - linas 20:48, 30 Apr 2005 (UTC)

I agree.-XaosBits

Butterfly

I don't know if the butterfly should really have its own page as it is only an example of chaos theory

  • well that's what most people talk about when they hear "chaos theory" you know the butterfly flaps its wings in africa and you get a typhoon in china or something like that... there's a film coming out called chaos theory soon i think, i'm sure that'll make this article very popular, it'd be nice to address some of the common conceptions and misconceptions...

Does anyone here besides me think that fractals deserve some mention in the article?

Fractals could indeed be mentioned if strange attractors where also mentioned. I'd do it myself if it weren't because someone else seems to be editing the page right now (concurrent editing is a pain, even under WikiWiki :).

Regarding the Lorenz attractor, take a pick on http://www.google.com/search?q=Lorenz+attractor.

--Filip Larsen

I think the statement "Strange attractors have a fractal-like structure." might be too strong. I don't think that is proven. --jkominek

Please tell me that real academics don't use the word "dynamical". The "ic" and "al" suffixes mean the same thing (they both mean "pertaining to"). It's just like nails on a chalkboard to me, especially considering that looking the word up in a dictinoary yields the definition "see dynamic".

Real academics might not use the word, but mathematicians certainly do. "Dynamical systems" is what it's called everywhere. --Axelboldt
Just looked it up in the OED. ::sigh:: stinking Greek word not adhering to the system of short roots everywhere else in english...

This Artical does also not mention using the logistic equation to generate (pseudo)random numbers, nor does the artical on pseudorandom numbers mentions chaos theory. Both of which are interesting, as numbers generated in such a manor would theoretically have no period.

I'm new, so I still don't know how does it work. Senzitivity on initial conditions and boundedness are not enough to get chaotic motion. They can imply for instance Quasi-periodicity. In 2D continuous dynamical systems you can have both conditions fullfiled and you"ll stil wont get a chaos. I also think that Entropy should necessary be mentioned on this page.

mention of knot theory as well?

Knot theory is more a branch of topology, not of chaos theory. AxelBoldt 22:13 Sep 26, 2002 (UTC)

This article seems to address the general issue of nonlinear dynamical systems, of which chaos theory forms a subset. I think it should be made VERY clear that the fields of nonlinear dynamics and chaos theory are NOT the same -- nonlinear dynamics deals with any dynamical system that displays nonlinear behavior. ONE of these types of systems is a chaotic system. However, a choatic system is a VERY SPECIFIC type of nonlinear system, it is a system that must satisfy a very specific set of properties, e.g. as set out in the original paper with that title something like "period 3 implies chaos" that started the whole thing. However, NOT every nonlinear system is chaotic, and confusing the two, implying that "chaos theory" and "nonlinear dynamics" are equivalent (which is a very common thing that people do) is not correct and gives a very misleading impression.

Yep, I throughly agree with this--I removed the 'also termed nonlinear dynamics' part to fix it up a bit, but the whole intro could do with a rewrite that I don't have the knowledge to do right now--have to wait until I have some textbooks beside me. —chopchopwhitey 00:40, 3 Feb 2004 (UTC)
OK I rewrote it slightly sans textbook. Could still do with a bit of work. —chopchopwhitey 00:49, 3 Feb 2004 (UTC)

The origins of chaos theory go much farther back than the 1950s. Similar problems were studied by Poincare, Kolmogorov, and others. Mathematicians were thinking about this long before experimentalists "discovered" it. The math was just very difficult to read and lacked intuition because the physics hadn't caught up.

As far as I know it was Poincare who first observed chaos when he was studying the three body problem. Why do they discuss hydrogen atom in Quantum Mechanics until it is beaten down to death, but do not mention anything about Helium? Helium atom is a three body problem displaying chaotic behaviour.

http://www.fsz.bme.hu/~pholmes/#jan12 --Akella 01:38, 5 Feb 2004 (UTC)

Removed this part

[Brazilian Portuguese Version]
Estabelecida em 1960, a Teoria do Caos lida com Sistemas Dinâmicos que, a princípio, são determinísticos, porém possuem uma enorme sensibilidade às condições iniciais devido a suas equações não-lineares. Exemplos desses tipos de sistemas são as placas tectônicas, atmosfera terrestre, economia e crescimento populacional.

The portuguese version belongs in the Portuguese Wikipedia. An interlanguage link might then be added into this article. Zubras 03:37, 23 Dec 2003 (UTC)

Would it be possible to add a section to the "Chaos Theory" article discussing the popular adoption of the theory, and the common misinterpretations that result? I'm thinking of Jurassic Park as an example.

-Mike, an interested reader

Sounds like a good idea. Also, the comment inserted by the anonymous poster is right. An ordinary reader would get the idea that linear models and linear mathematics was dead as dust and useless for most circumstances, which is hardly the case. Revolver 14:01, 2 Nov 2004 (UTC)
I started responding to that bit, and just now read this conversation. I'm not a physicist, but I have some familiarity with non-chaotic physics. And as a SF fan, I've seen the term chaos-theory used in lots of inappropriate places (any mysterious invention, usually). I'll have to watch Jurassic Park again, though. I put in a section called Popular Conceptions, which I'm working on, but any help on the text or title would be appreciated. By the way, does anybody know where the elephant's tail saying came from? WhiteC 17:26, 8 Nov 2004 (UTC)
For obvious reasons most man-made systems are intentionally designed to be both linear and stable (although see relaxed stability for an exception), so linear mathematics is far from dead. However, the study of non-linear systems is equally important because (a) systems designed to be linear may become non-linear and unstable in certain conditions (for example, see Tacoma Narrows Bridge and London Millennium Bridge) and (b) complex systems, both natural and man-made, will almost certainly behave in a non-linear fashion, so non-linear dynamics is important in fields from meteorology to economics. Not sure how this fits under Popular Conceptions, though. Gandalf61 15:30, Nov 9, 2004 (UTC)
I have updated the article to incorporate some points from this discussion. Gandalf61 12:05, Nov 11, 2004 (UTC)
Thanks. I am trying to think of a way to do this Popular Conceptions without it just being a refutation of the two bullet points. Or at least, a better way of wording it.WhiteC 19:06, 12 Nov 2004 (UTC)
I am thinking of putting in something about Brownian Motion. As I understand it, brownian motion of an enclosed system (say a tank full of gas) will include huge numbers of linear systems--molecules colliding, which can be aggregated statistically in a Stochastic process, but chaos theory is not involved (although the everyday meaning of 'chaos' would apply). Am I correct? Could a physicist help me out here? WhiteC 22:04, 4 Dec 2004 (UTC)
Would the random motion of an individual molecule be a chaotic thing or not? I guess that is the heart of the question. WhiteC 22:26, 4 Dec 2004 (UTC)

I'm not too good with maths, got interested in chaos theory by chance, I wonder what 'deterministic' means in Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense? Does this just mean that there are strict laws that govern the behaviour of the system? The link goes to a disambig page which links to philosophical determinism and to an article about algorithms (I think that it doesn't mention deterministic algorithms, though). There's also an article called Scientific determinism.. and none of those articles really help! Am I too stupid for this or is there something lacking in those articles about deterministic things? -- Jashiin 17:15, 1 Dec 2004 (UTC)

'Deterministic' just means that they follow the laws of cause and effect; cause Determines effect. There aren't any mysterious miracles or totally random things going on. In chaotic systems, it can be difficult to find out what the causes are for any given effect (making the system appear 'chaotic' in the everyday sense of the word), but that doesn't mean that there isn't a cause. I hope that helped. (Perhaps one of these articles needs a better introduction.)WhiteC 20:51, 1 Dec 2004 (UTC)
Ah, so thats what it is! Thank you very much :) As for these articles, maybe a little separate article is needed, something like Deterministic (mathematics).. -- Jashiin 20:43, 2 Dec 2004 (UTC)
I cross-posted this discussion to Talk:deterministic universe, which I think should be changed to 'deterministic system' (which does not yet exist). I tried looking through the disambig page and I agree it is quite difficult to figure out exactly what is meant by 'determinstic' from there. WhiteC 21:04, 3 Dec 2004 (UTC)

Invitation

Work is currently in progress on a page entitled Views of Creationists and mainstream scientists compared. Also currently being worked upon is Wikipedia: NPOV (Comparison of views in science) giving guidelines for this type of page. It is meant to be a set of guidelines for NPOV in this setting. People knowledgable in many areas of science and the philosophy of science are greatly needed here. And all are needed to ensure the guidelines correctly represent NPOV in this setting.  :) Barnaby dawson 21:10, 29 Dec 2004 (UTC)

Picture of fractals

I think it would be nice if there was a picture of a suitable fractal or something that illustrated an example of chaos theory. Does anyone know anything suitable before I start looking? WhiteC 20:51, 6 Mar 2005 (UTC)

I would hate to see pictures of fractals appear here without an appropriate discussion of the connections between fractals and chaotic dynamics. I think there are already too many books and articles out there that talk about chaos theory (either from a technical or more broadly accessible standpoint) and then just slap a pretty but inappropriate picture of a fractal on the cover without explaining why. I'm not saying it wouldn't be nice to have pictures of fractals here. I'm just suggesting we should hold off with pictures until it actually matches the text. CyborgTosser (Only half the battle) 22:29, 4 May 2005 (UTC)

There are many free programs on the web which will generate pictures of fractals. I don't know of any pictures already uploaded to wikipedia though. Barnaby dawson 08:45, 7 Mar 2005 (UTC)

Criticisms of chaos theory

I have a number of issues with the first paragraph here, and I think it needs major work. I will probably come back and do it myself, but I am bogged down right now and maybe someone else with knowledge in this area can get to it before I do.

Most signals used in signal theory have finite energy and are linear.

Signals are linear? I have no idea what this is supposed to mean. And signals having finite energy has nothing to do with the possibility or otherwise of chaos, not to mention that it's not true (in communications theory it is far more common to deal with signals that have finite power rather than finite energy)

This is a requirement in order for the majority of study in signal theory to be valid.

No, it is a requirement to use linear tools for signal analysis and processing.

Control theory is also linear.

There is linear control theory and non-linear control theory.

Mechanics, Dynamics and Statics are all based on linear equations that describe the world. Such equations are the basis for the simulations used to launch satellites into space, build solid bridges, etc.

I would like to see anyone launch a satellite into space using only linear differential equations. Most of what you learn in an introductory statics or dynamics class will deal with linear equations, but nonlinearity is far from irrelevant to engineering. In fact, before I saw it here, I had never seen this brought up as a criticism to chaos theory. CyborgTosser (Only half the battle) 23:05, 4 May 2005 (UTC)

Quite right. I removed the entire section and pasted it below. Transistors are insanely non-linear, yet we engineer with them all the time. linas 00:59, 10 May 2005 (UTC)

==Criticisms of chaos theory==
One criticism of chaos theory is that it focuses on behavior that is of peripheral importance in real-world engineering. For example, passive electrical circuits are linear. Most signals used in signal theory have finite energy and are linear. This is a requirement in order for the majority of study in signal theory to be valid. Control theory is also linear. Mechanics, Dynamics and Statics are all based on linear equations that describe the world. Such equations are the basis for the simulations used to launch satellites into space, build solid bridges, etc. These are all important real-world systems.
A response to this criticism is that the study of non-linear behavior is still relevant to the design of linear systems because even systems designed to be linear and stable may become non-linear and unstable in certain conditions (see Tacoma Narrows Bridge and London Millennium Bridge).
Another response to this criticism is that engineers have selectively chosen to work with linear systems wherever possible because they are easier to understand.

Determinism and chaos

I am somewhat confused by the following claim:

Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense ..

Does this preclude the existence of random fractals like:

  • randomly populated 2 dimensional lattices that model trees in a forest that is susceptible to forest fire, or
  • any other phenomenon involving the notion of percolation threshold?

I am sure my confusion is over terminology as in the distinction between non-determinism and randomness. Yet this claim was made in the introductory paragraph that was intended for a lay audience, so I might not be the only one who is confused.

Thanks ahead of time for anyone's help Vonkje 03:18, 1 Jun 2005 (UTC)

The sentence is trying to highlight the fact that the mathematical systems studied in the field of mathematics called chaos theory are actually deterministic systems, not systems with random (or pseudo-random) elements. Randomly populated lattices, for example, would be studied within a field such as stochastic processes or, more specifically, percolation theory, but not within chaos theory. No one is surprised that mathematical systems with random elements have a high degree of sensitivity to initial conditions - they are specifically designed to exhibit this behaviour. What is surprising is that quite simple deterministic systems can also show this sensitivity - these are the types of systems that are studied in chaos theory. Gandalf61 09:34, Jun 1, 2005 (UTC)
Gandalf61, would you care to take what you just said above, boil it down to one sentence, and add it to the article? i.e. that chaos theory is not about non-deterministic fractal theories. linas
Thank you Gandalf61. You have been most helpful. My confusion lies in misunderstanding the intention of the subject statement. From what I now gather, the statement was intended to delineate the scope of Chaos theory, lest it be further conflated by popular culture (ie: chaos washing machines???). Considering Linas' suggestion, could a suitable replacement for the subject statement:
Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense ..
... go something like this:
It is surprising that Chaos theory confines its study to deterministic systems, since these systems' behavior can be so complex. Vonkje 04:57, 3 Jun 2005 (UTC)

Hmmmm. What is surprising is that determinism is chaotic. It is not surprising that chaos theory confines its study to deterministic systems. No, I was rather thinking that a sentence should be added that says something like Chaos theory is limited to the study of deterministic dynamical systems. Many non-deterministic systems also exhibit fractal behaviour; this broader field of study is called "non-linear studies". linas 05:24, 3 Jun 2005 (UTC)

I like the clarity of the paragraph from the article on Dynamical system at the beginning of the section 'Dynamical Systems and Chaos Theory'
Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. (Remember that we are speaking of completely deterministic systems!). This unpredictable behaviour has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.
Perhaps the bit in parentheses could go, but I think it sums things up fairly well. Just put in the link back to dynamical systems, and you're set. WhiteC 08:19, 3 Jun 2005 (UTC)
It's sounding better and better. ... I am inclined to agree with Linas' observation that what is surprising is that determinism can be chaotic. WhiteC's quote above works for me providing that determinism can be associated with chaos, and that all chaotic systems are deterministic. Mention of determinism in this context twarts two mistaken public perceptions that undervalues determinism while conflating chaos theory (think for a moment of high school students doing term papers here): (a) "Determinism is simple and thus uninteresting" (wrong), and (b) "Chaos is complicated so I should believe what I read" (wrong). Vonkje 13:16, 3 Jun 2005 (UTC)
The ordinary non-technical usage of the word chaotic in the English language will frequently cause problems and/or surprise here. I think the differences between that and chaos theory (which IS deterministic, although in practice it may be difficult to predict a chaotic system's behavior) should be made as explicit as possible. I think the technical term chaos-theory and common-usage of 'chaos' can be different. This difference should be mentioned in the first sentence of this article. The popular conceptions part started when I saw how often these kind of mistakes happen. WhiteC 03:20, 4 Jun 2005 (UTC)

Alternating Paradigms

Moved following speculative sections here from main article. Gandalf61 08:43, August 1, 2005 (UTC)

Surprisingly, both paradigms (discrete and continuous) seem to represent surprisingly well various aspects of reality, without undue strain on the models presented. The question arises, then, is reality fractal, or is it chaotic?

This can also be expressed as whether reality functions in a continuous manner, or in a discrete manner, thereby defining which branch of mathematics is more important (at least, to the applied mathematician). It may be, without providing a strong proof or an exhaustive argument at this point, that reality alternates between both paradigms, and at times is discrete (fractal), and at times continuos (chaotic). For instance, if we begin at the smallest scale, the quantum scale, we will see that quantum states are discrete, and may be interpreted later on by fractal geometry. As we proceed to our everyday world, we notice it is mostly continuous, and hence chaotis. This is seen in the study of continuous motion, from a physicist's point of view. Finally, we return to a fractal framework when we regard Einstein's theories of special and general relativity, which both provide a frame of reference that can also be seen as quasi-discrete, and may be interpreted by future physicists and being inherently fractal in nature. At this point, we are merely speculating.

billiards

What is dynamical billiards? Whenever I've played, it is always called billiards, pool or snooker. The billiards article makes no mention of 'dynamical'. Having such a link is confusing--are you trying to make some sort of point about billiards itself, or its associations with determinism/chaos theory? I suppose you could use billiards to illustrate chaotic/dynamic systems if you wanted to, but it seems confusing to do so when billiards was often used as an example of a classical deterministic system. WhiteC 17:59, 9 August 2005 (UTC)

I think this means Sinai's billiards or Lorentz gas. Its a mathematical concept of bouncing a point particle around on an n-dimensional surface (often a torus). See my web page Sinai's Billiards. BTW, note that page and its images are under GFDL. linas 00:27, 10 August 2005 (UTC)
Ah, that makes sense. It is the link to the regular billiards article that confused me. WhiteC 03:37, 10 August 2005 (UTC)

AUTODYN

I removed AUTODYN. It is a commercial package for solving complicated non-linear PDEs. If simulation packages are going to be mentioned in the See also section, there are several other packages that are more in line with the content of the article. XaosBits 02:24, 23 August 2005 (UTC)

Yes, right, it doesn't belong here. I think there is a category devoted to math software somewhere. linas 04:26, 23 August 2005 (UTC)

Removed incorrect paragraph

An anonymous user added the following paragraph:

In layman’s terms, chaotic motion is akin to a dynamic system having a case where a single initial condition has more than one solution possible, so that there is no inherent method of selecting which feasible solution will be realized. Usually the two solutions from the same starting point are radically different from each other. Infinitesimal differences in initial conditions or parameters cause one solution to be selected over the other. The Chaos region would be a portion of the state space (trajectory positions and velocities) representing unique initial conditions that posses non-unique solutions.

I reverted this change because this is paragraph is incorrect in two ways. First of all a dynamic system cannot have more than one solution for the same initial conditions - what it can do is exhibit very different behaviours for two sets of initial conditions that are close to one another (perhaps arbitrarily close), but still different from one another. Secondly sensitive dependence on initial conditions is not a sufficient condition for chaos on its own - a pencil balanced on its end is very sensitive to initial conditions, but its final behaviour (lying on its side) is a stable fixed point, not chaotic. The additional conditions of transitivity and dense orbits are also necessary in any reasonable definition of chaotic behaviour. Gandalf61 10:05, 25 November 2005 (UTC)

Heavy reading

Greetings all. I'd just like to take a moment to mention that this article is pretty freaking heavy reading, and that's perfectly fine, but it would not hurt to maybe think about some ways to make the article a little more accessible. Now, I don't know diddly squat about chaos theory, so I'm gonna spend some time trying to see if I can work the prose around into something that is a bit more readable, and I'd welcome input to make sure I don't introduce anything innaccurate in so doing. I'll be asking here for help on clarifying anything I think needs more explanation. Cheers! Ëvilphoenix Burn! 07:12, 20 December 2005 (UTC)

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