# Talk:Chaos theory/Archive 6

 Archive 1 ← Archive 4 Archive 5 Archive 6

## Phase diagrams

"Phase diagram for a damped driven pendulum, with double period motion" is of poor quality. The plot is not doubly periodic (possibly due to a large integration step). Digitalslice 13:51, 4 June 2007 (UTC)

From the description, this plot seems to be generated from experimental data rather than a numerical simulation - thus the inexact measurements. Chrisjohnson (talk) 01:52, 30 January 2008 (UTC)

This is beneficial to know. Chaos has no laws, has no form, has no gender, chaos exists as a prime function of our existence to guide towards the future.-Z —Preceding unsigned comment added by 96.251.128.33 (talk) 06:05, 27 February 2010 (UTC)

## The relation between chaos theory, systems and systems theory

The importance rate of this article for the WikiProject Systems has been uprated from high to top allready two weeks ago on 10 June 2007. I could have referted it because importance rates are set by the WikiProjects themselves and these rates have a particular objective meaning: The importance rate is not about the objective importance of the article, but of the relative difference from the article to the hart of the WikiProject. Now formaly the hart of the WikiProject Systems is in a way the category:systems. The items in this category get a top-importance, the items in the first subcategories are of high-importance.

Instead of referting this I kept wondering about the relation between chaos theory and systems and systems theory. Is or isn't chaos theory in the first place about chaos and not about systems. And aren't systems in the first place about organization and not about chaos? I know a bit more about systems theory, a little about chaos theory but even less about the role of systems and systems theory in chaos theory. Can somebody explain this to me? - Mdd 19:41, 25 June 2007 (UTC)

Chaos is the origin: the original god that we are aware of, even though chaos has origins further removed. —Preceding unsigned comment added by 96.251.128.33 (talk) 06:07, 27 February 2010 (UTC)

I can't really answer the question, and would encourage others to do so, because it is an interesting and important question, but I should make some comments about changing the ratings.
• First of all, I shouldn't have changed the systems theory importance rating, because this is the importance of the article for WikiProject Systems, and I don't know how that project assigns these ratings. Please change it back to "High" if you think it is appropriate: different projects can of course have different ratings for the same article.
• Second, some background. At the Mathematics WikiProject, we are finding that too many articles are getting Mid and High importance ratings compared to Top and Low. In particular, this makes it harder to prioritise which Stub and Start-Class articles at the top end most need expansion. So we have been trying to improve the situation, and have developed more detailed importance criteria to help us.
• Third, my changes here. I uprated the Mathematics importance from High to Top (by the above reasoning). Now, WikiProject physics is rather inactive right now, and I figured this article is at least as important in physics as maths, so uprated the physics importance too. Then I went a bit far by thinking "Well, if it is top for maths and physics, it probably is for systems too"!
• Fourth, a comment. From what you have said, I understand that WikiProject Systems assesses importance in an absolute sense, i.e., only the main items in Category:Systems can hope to be top priority and so on. We discussed this quite a lot at the mathematics project, and have come to the conclusion that:
• it is more helpful to assess the importance of an article within context rather than in absolute terms
• Wikipedia 1.0 actually recommends this approach.
Now I am very impressed that your response to my mistake was not to revert it, but to think about it and raise such interesting questions. Maybe you might want to take some of the maths project thoughts on importance ratings back to WikiProject Systems and initiate a debate. All the best, anyway. Geometry guy 20:19, 25 June 2007 (UTC)

I will answer the questions refering to the assesment of article further on the talk page of the WikiProject Systems. And I would like to leave my question about the relation between chaos theory and systems and systems theory here for others to respond. So if anybody can help me out? - Mdd 22:22, 25 June 2007 (UTC)

## The role of systems theory in chaos theory

I am not sure that systems and systems theory can be said to have a role in chaos theory. I think it is rather the other way around; chaos theory has a role in systems and systems theory (from chaos emerges order and/or a system). In economics, notably, this is exposed through the concept of spontaneous order. See also Complex system#Complexity and chaos theory which has some info, although probably not perfect. --Childhood's End 13:56, 26 June 2007 (UTC)

Thanks for this perspective. This brings me all kind of questions. Is chaos theory a new paradigm in the field of systems theory. Should you in the first place name that field systems theory? Did or didn't the chaos scientists thought that chaos theory was a field on it's own. What did they think about the relation to systems theory? Now I am going to read the parts you mentioned and probably come up with questions? We'll see? - Mdd 14:58, 26 June 2007 (UTC)
I'm still wondering about the question is chaos theory can be seen as a form of systems theory? I found only partly an answer in a discussion here from March 2006, see [1]. - Mdd 00:12, 27 June 2007 (UTC)
hi mdd, my below post was partly in response to you. as described in the chaos theory wiki and in the dynamical systems wiki chaos theory is a fairly well defined island of mathematics. It has it's own language and its own set of tools used to get results. As such it is a good LANGUAGE and TOOL that can help discussion of systems theory (what ever THAT grab bag might be). Notice in the talk page for dynamical systems they are choosing to include only systems acting on what is called a smooth mathematical space, and therefore leaving out such topics as (discrete) cellular automata and networks etc... Again, this makes that chunk of tools very specific, these mathematicians have developed many tools that work on giving results in these smooth systems, but DON'T KNOW yet how to get results in the discrete systems. Leaving again, complex systems, general systems, emergence... to be more general more varied topics.
So, not every complex system is approachable yet by the mathematical tools of dynamical systems theory, and remember, not every complex system exhibits chaos.
One more point: chaos theory and the more general dynamical systems theory are deterministic systems, they do NOT involve chance or random input. That is yet a whole other body of mathematics! Many of the systems under the topics of complex systems and general systems, i presume, include random elements. they require other tools.
Certainly SOME of the systems explored in systems theory and complex systems have served as inspiration to people developing the mathematical results of chaotic dynamical systems, but they necessarily have to choose very simplified examples in order to do their work.
Remember that most of this stuff has only been developed in earnest in the last 50 years! It is uncharted territory, still in flux. that is why i still stand by my conclusion that ALL OF THIS might best be approached for an encyclopedia as a set of very separate topics each with their own approach and insight and let the reader make his own connections. Otherwise we will end up in very strongly point of viewed personal ramblings, as i have done in my attempts to bring this stuff together in my own mind these past 20 years, resulting in my decision for my own writings to write 60 separate lab manual entries for each kind of system/topic.
however this is exciting that you guys are attempting this and i will mull this all over in the coming weeks.Wikiskimmer 19:37, 29 June 2007 (UTC)

## Bleach on the term chaos theory - what a nest of hornets

bleach on the term chaos theory! the concepts described in this wiki are basically a solid body of well defined MATHEMATICS. It describes a subset of the area of mathematics called dynamical systems. as such it is an excellent tool for some other more complicated human endevours like complexity, systems theory etc... as a body of mathematics it stands on its own two feet.

i've just started looking at all these related wikis. my god. what a nest of hornets! Wikiskimmer 05:40, 29 June 2007 (UTC)

And again, in English? -- GWO
the name chaos theory sounds too mush brains. i think the term used by mathematicians is chaotic dynamical systems. Wikiskimmer 19:39, 29 June 2007 (UTC)
What about these 1500 books that call it "chaos theory"? Dicklyon 20:12, 29 June 2007 (UTC)
The body of this wiki fairly clearly discusses the specific body of mathematical work on chaotic dynamical systems. perhaps a mention at the beginning can be made of the broader usages of the term in science, engineering and pop culture. i am exploring the tangle that all the wiki articles related to 'systems' is in. i think in an encyclopedia, the less tangle the better. Wikiskimmer 21:34, 29 June 2007 (UTC)

### why not just Chaos?

There's not really a formal "chaos theory" as Wikiskimmer is saying. Chaos is a special case of dynamics in dynamical systems theory Xurtio (talk) 10:15, 10 August 2010 (UTC)

Firstly, the term "Chaos theory" is widely used (see elsewhere on this page), and secondly, the name Chaos was already taken. -- Radagast3 (talk) 10:55, 10 August 2010 (UTC)
1) Yeah, but this article seems consistent with the formalism used by dynamical scientists. I'm a student of dynamical science so I don't have ultimate authority, but I believe Wikiskimmer's complaints are legit. 2) disambiguation page: Chaos (dynamical systems). Xurtio (talk) 11:17, 10 August 2010 (UTC)
Well, I would note that those comments were made in 2007, and nobody else has supported them. For the record, I oppose the suggested page move. -- Radagast3 (talk) 11:32, 10 August 2010 (UTC)
Understood. I'll make my case and then let it go to the community: Google 'chaos theory' and watch the plethora of unsubstantiated crap pile up. Go to google scholar and do the same thing: you'll see no nonlinear dynamics papers... just outside fields. Google scholar: 'chaos' and you'll see all the dynamical papers you'd need to write this Wiki article. Unless somebody makes a silly argument, I won't push the subject anymore. Xurtio (talk) 11:37, 10 August 2010 (UTC)
I just wanted to add that there is no "chaos theory" (I do research in chaos by the way, as a grad student). Anyway, there's no actual "theory" in the scientific sense. Chaos is a phenomenon, like weather, space, or time. That's the issue. But the term is still used (even by professionals) so it's not exactly out of place either, so I offer a sort of compromise: Add a section to this article "Not a theory" (I can find sources for this, it's been discussed before. Will post them after I do some research and validity, etc). The same discussion has come up for "systems theory" too. Xurtio (talk) 08:50, 30 August 2010 (UTC)

## to redirect

Mdd, if you put brackets around chaotic dynamical systems then make the thing redirect to chaos theory because THAT article IS chaotic dynamical systems. I don't know how to redirect.Wikiskimmer 05:47, 3 July 2007 (UTC)

To redirect you just click on the red-lighted word chaotic dynamical systems, then a new page starts. In the text field you then write #REDIRECT [[chaos theory]]. Next time you see the chaotic dynamical systems it's turned blue. You then created a new page: a redirect page I call them. A good thing is to search in the Wikipedia for this word in articles and there putt brackets around them. Better you do this before you make a redirection page.
Now I did some searching for you and found that in the article Floris Takens the term chaotic dynamical systems is redirected in an other way, like chaotic dynamical systems or in plain nowiki-text [[chaos theory|chaotic dynamical systems]]. It all that's some time getting use to. You should just try different times. Good luck with it. - Mdd 11:38, 3 July 2007 (UTC)
ok, i did that. but here's another question. can i do it the other way around? can i change the name of the chaos theory article to "chaotic dynamical systems" and have chaos theory redirect to IT? That would be a minor esthetic improvement, as in english the term "chaos theory" still sounds to wishy washy, a theory of (general) "chaos", as in what my bedroom looks like, while "chaotic dynamical systems" refers to the mathematically defined systems that exhibit "mathematically defined chaos".Wikiskimmer 14:14, 3 July 2007 (UTC)

### Moving chaos theory to chaotic dynamical systems ??

Wikiskimmer, if I understand you correctly, you are proposing to move chaos theory to chaotic dynamical systems. I see two problems with this. Firstly, a small problem - WP:NAME says "In general only create page titles that are in the singular", so the new name would have to be chaotic dynamical system. Secondly, a bigger problem - WP:NAME also says "Except where other accepted Wikipedia naming conventions give a different indication, use the most common name of a person or thing that does not conflict with the names of other people or things". Like it or not, chaos theory is a more common name for the subject of this article than chaotic dynamical systems. Anyway, before you change anything, I suggest you mention your proposed name change at Wikipedia talk:WikiProject Mathematics and see what the general reaction is. Gandalf61 16:12, 3 July 2007 (UTC)

if the goal is to be a POPULAR encyclopedia instead of a MATHEMATICAL encyclopedia, then i suppose chaos theory might be the most popular term. but the popular notion probably points to a broader category than the math that's in our chaos theory article. and all of a sudden i'm wondering, in the grand scheme of things, just how important is this anyway?Wikiskimmer 16:41, 3 July 2007 (UTC)
I oppose such a move. The topic is widely known as chaos theory, and there's no reason to mess with it. Dicklyon 00:08, 4 July 2007 (UTC)

Today 16 july 2007 I made a rather large reorganization of this article. The main idea behind it is:

1. In the first place an introduction as it was.
2. Second an part about the history
3. And in a third part all theoretical parts together
4. And the ending with references is reorganized according to Wikipedia standaards

I hereby kind of followed the example of the featured article Electrical engineering. Following this example also gives an idea how this article can be further improved. It looks to me that improvements can be made o points like: Education, Practicing & Applications!? - Mdd 12:50, 16 July 2007 (UTC)

"Chaos" is a tricky thing to define. In fact, it is much easier to list properties that a system described as "chaotic" has rather than to give a precise definition of chaos. Gleick (1988, p. 306) notes that "No one [of the chaos scientists he interviewed] could quite agree on [a definition of] the word itself," and so instead gives descriptions from a number of practitioners in the field. For example, he quotes Philip Holmes (apparently defining "chaotic") as, "The complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems." Similarly, he quotes Bai-Lin Hao describing chaos (roughly) as "a kind of order without periodicity." It turns out that even textbooks devoted to chaos do not really define the term. For example, Wiggins (1990, p. 437) says, "A dynamical system displaying sensitive dependence on initial conditions on a closed invariant set (which consists of more than one orbit) will be called chaotic." Tabor (1989, p. 34) says, "By a chaotic solution to a deterministic equation we mean a solution whose outcome is very sensitive to initial conditions (i.e., small changes in initial conditions lead to great differences in outcome) and whose evolution through phase space appears to be quite random." Finally, Rasband (1990, p. 1) says, "The very use of the word 'chaos' implies some observation of a system, perhaps through measurement, and that these observations or measurements vary unpredictably. We often say observations are chaotic when there is no discernible regularity or order." So a simple, if slightly imprecise, way of describing chaos is "chaotic systems are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random." In particular, a chaotic dynamical system is generally characterized by 1. Having a dense collection of points with periodic orbits, 2. Being sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), a property sometimes known as the butterfly effect, and 3. Being topologically transitive. However, it should be noted that despite its "random" appearance, chaos is a deterministic evolution. In addition, there are chaotic systems that do not have periodic orbits (periodic orbits only survive in the boundaries of KAM tori, and for sufficiently strong perturbations from the integrable case, islands do not necessarily survive). Furthermore, in so-called quantum chaos, trajectories do not diverge exponentially because they are constrained by the fact that the entire evolution must be unitary. The boundary between regular and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Strogatz 1994, pp. 363-365). An example of a simple physical system which displays chaotic behavior is the motion of a magnetic pendulum over a plane containing two or more attractive magnets. The magnet over which the pendulum ultimately comes to rest (due to frictional damping) is highly dependent on the starting position and velocity of the pendulum (Dickau). Another such system is a double pendulum (a pendulum with another pendulum attached to its end). M. Tabor and F. Calogero have advocated an interpretation of chaos as motion on Riemann surfaces (Tabor and Weiss 1981, Fournier et al. 1988, Bountis et al. 1993, Bountis 1995).

## its periodic orbits must be dense.

I think that's about where my tigerdilly should go. It's an inversion of an escape-time fractal (a complication of the Mandelbrot Set), so a large area of periodic orbits is in it, but the sparse, textured area of escapes finds difficulty in analysis. Brewhaha@edmc.net 18:45, 26 August 2007 (UTC)

## A essay about chaos theory

Tonight User:71.185.153.98 dumped an essay "Chaos Theory: A Brief Introduction" into this article. I for the moment moved it back to User talk:71.185.153.98 page. Maybe someone wants to take a look at it. - Mdd 20:13, 9 October 2007 (UTC)

## Infinity and Circular views

Hi, I am very new here. This is the first time I do this. I hope I don't mess anything up. I have read this passage in the article.

"Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behaviour, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems."

I understand it completely until the part where it talks about using bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle. I think I have an idea of what it means but it seems rather far fetched to me and I woul much rather hear some explanations before looking like a fool. What is the difference between bounded and unbounded metrics? I wish to get some clarification on this as well.

Again, sorry if I did this wrong and if I didn't do it wrong then thanks for your input!

Alkaroth 11:41, 18 October 2007 (UTC)

I will attempt an explanation. Each point on a unit circle C is defined by an angle θ, which we take to be in the interval -π < θ <= π. We can map the real line R to the circle C in various ways. One way is to define a function θ:R->C such that θ(x)=2tan-1(x). This is an bijection between R and the subset -π < θ < π of C. It is also a continuous function (although it is not uniformly continuous). The only point on C that is not an image of a point in R is the point θ=π, that is "opposite" to 0. If we add a "point at infinity" to R with the convention that tan-1(infinity) = π/2 then we have a bijection between R+{infinity} and C. With this mapping, C is called the real projective line.
The dynamical system x->2x is a dynamical system on R. But if we map it from R to C then the behaviour of every point (except for the fixed point 0) is identical - they all converge to the point θ=π. So this dynamical system is clearly not "chaotic" on C. As we have used a continuous mapping, we can reasonably argue that we have not changed any fundamental property of the dynamical system by this mapping - and we would like "chaotic" to be a fundamental property that is not changed by a continuous transformation. So the dynamical system x->2x is not usually described as being "chaotic", even though it could be said to exhbit "sensitivity to initial conditions" when considered as a dynamical system on R. Gandalf61 12:57, 18 October 2007 (UTC)
Gandalf, Thanks a bunch for the explanation. I understood it better and it was similar to what I had in mind. Thanks again for your help Gandalf. Alkaroth 13:31, 18 October 2007 (UTC)

## Chaos analysis software

It would be nice to explain in the article which software tools or languages are available for the analysis of chaotic systems. —Preceding unsigned comment added by 83.34.43.235 (talk) 18:55, 11 November 2007 (UTC)

Finally I find a nice software to study chaos: TISEAN. —Preceding unsigned comment added by 81.35.123.125 (talk) 17:56, 24 April 2008 (UTC)

## Distinguishing random from chaotic data

I'd like to add another reference Physics Letters A, Volume 210, Issues 4-5, 15 January 1996, Pages 290-300 Reconstructing the state space of continuous time chaotic systems using power spectra J. M. Lipton and K. P. Dabke (at the very end of this section) but as a co-author I'm conflicted. Anyone keen to confirm this as a reasonable citation and add it? Thanks Jmlipton (talk) 05:14, 18 December 2007 (UTC)

## A section a simpler terms?

Could it be possible to add a small section that pretty much stated the chaos theory in 'plain English'? Stepshep (talk) 02:07, 8 December 2007 (UTC)

The first sentence of the article has a high-level informal definition of chaos theory: "... chaos theory describes the behavior of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect)". Follow the links to find out more about dynamical systems and the butterfly effect. There is an expanded introduction in the Simple English wiki here. But if you want to know exactly what chaos theory is about (i.e. what exactly is known about the behaviour of chaotic systems and how do we know it) then you have to understand some of the mathematics behind it. Gandalf61 (talk) 12:24, 9 December 2007 (UTC)

I agree with Stepshep. Whoever wrote this article was high on chaos-related jargon and low on writing skills and communication ability. For the purposes of your FIRST SENTENCE, you should probably avoid multiple other articles that must be read. That becomes a slippery slope (i.e. what if those articles also require more reading) that makes encyclopedias useless.

Example...the phrase "certain nonlinear dynamical systems" can be simplified to just "certain systems." It means the same thing, and you can add exactly what type of systems later. Likewise, you can replace "may exhibit dynamics that are highly sensitive" with "may be highly sensitive." You don't need the excess detail, especially in the first sentence which is supposed to plainly state what you're discussing.69.232.97.150 (talk) 02:49, 6 February 2008 (UTC)

## Possible consequences

The way, I've always been told the chaos theory, is when a butterfly flaps its wings, disaster strikes. Obviously, these are pretty extreme circumstances, and I am currently unaware of the full details, but if anyone could possibly let me know, about the reprimands of chaos theory, I would be very grateful. —Preceding unsigned comment added by Hammerandclaw (talkcontribs) 21:17, 29 December 2007 (UTC)

It is not that bad, fortunately. Most of the time, when a butterfly flaps its wings, no disaster strikes. The idea is, rather, that the effect is unpredictable, and that we cannot fully 100% guarantee that such a tiny and seemingly insignificant thing cannot lead to a large-scale effect, which, if we are unlucky, might be disaster. This does not only apply to butterflies flapping their wings right now, but also to someone scratching their nose, and the large-scale results we see are the combined effect of many such things over long periods, including Julius Caesar scratching his nose, and all flapping of wings by Jurassic butterflies 200 million years ago; each and any of that may be the difference between rain or sunshine tomorrow. See also Butterfly effect.  --Lambiam 23:28, 29 December 2007 (UTC)

## Chaos in Every day life

What about how chaos underpins such subjects as sociology, biology, politics, etc? 194.73.99.107 (talk) 11:31, 12 January 2008 (UTC)

Underpins? Sounds like someone's fanciful imagination. Dicklyon (talk) 17:44, 12 January 2008 (UTC)
Actually, it's not so fanciful. There have been studies into chaos in the behavior of stock markets, see also here. Biological systems are complex systems which have been described as "anti-chaotic" (see Stuart Kauffman's work, for example). The presence or absence of chaotic heartbeats have been implicated in human health (here and here). SteveChervitzTrutane (talk) 07:11, 9 April 2009 (UTC)

## Quick suggestions

The lead and general introduction should perhaps provide more concrete & practical examples of the so-called "Butterfly effect" (pendullum, etc.) History of discovery of this new "paradigm" should be more developed (here it seems everything is brought back to the "first discoverer of chaos", and then comes the computer... James Gleick's book might be of some help here in retracing the various discoveries and time needed to take them together). Technical information should come last. Right now the article is at the same time too short and too complex to provide a useful introduction to a reader totally unfamiliar with the subject (simple example: sensitivity to initial conditions & Butterfly effect is easy to understand for anyone familiar to this theory, but should be explained better here. An example from population dynamics could comes in handy (low fertility: extinction; medium fertility; regular increase; high fertility=phase 3 implies chaos...) Mandelbrot sets, fractals and the creative dimension in some fractals should also be depicted. Difference between chaotic & stable systems with non-chaotic stable systems should be explained. Lapaz (talk) 13:56, 17 January 2008 (UTC)

## Removed "philosophical" paragraph

I removed the following paragraph from the History section of the article:

"Philosophically, Chaos theory demonstrated that Laplace's demon deterministic assumptions were erroneous, as various outcomes could originate from the same initial situation. Furthermore, it showed the possibility of self-organizational systems, thus defying the second law of thermodynamics of increasing entropy. Chaos theory did not, however, reject all forms of determinism, but only Laplacian or classical determinism, which assumed that if one knew perfectly all of the coordinates of the universe at one point of time, one could predict all its past and future history. To the contrary, Chaos theory showed that if emergent properties arose from disorder and non-linear systems, thus creating novelty and dismissing the Laplacian hypothesis, the appearance of disorder itself and of non-regularity could themselves be predicted, in particular by using iterated function systems."

I believe this paragraph is incorrect. Firstly, chaos theory studies deterministic systems, so a given initial situation can only give rise to one outcome at a given later time. Laplace's demon could happily predict the behaviour of a chaotic system as long as it had exact knowledge of the initial conditions. What prevents Laplace's demon predicting the behaviour of actual physical systems is the inherently non-deterministic nature of quantum physics - but this aspect of reality is not studied by chaos theory, which only considers classical deterministic systems (except in the rather separate and specialised field of quantum chaos). Secondly, self-organizational systems do not defy the second law of thermodynamics. They are either open systems which decrease their local entropy by exporting entropy to the surrounding environment (typically by cooling, and so heating their environment), or they are closed systems which are initially prepared in a very specific state, and so have an extremely low initial entropy anyway. This is discussed in Self-organization#Self-organization vs. entropy. Gandalf61 (talk) 11:03, 18 January 2008 (UTC)

I agree it was too quickly formulated. Perhaps another formulation could be given to it, namely the distinction between determinism (maintained by chaos theory) and previsibility. Various views appears to spring up here: Jean Bricmont, for example, alleges that Laplace did not claim that determinism implied previsibility [2]. On the other hand, Bernard Piettre, director of studies at the College International de Philosophie, maintains exactly the reverse (link lamentably in French, maybe an automatic translator could work). Whatever the way, I think the philosophical issue should be adressed, and if various point of views supported, these one given. Maybe you have some other, not too technical, sources in English concerning philosophical implications of this chaos theory (which, we agree, is deterministic)? Lapaz (talk) 19:42, 23 January 2008 (UTC)
Stephen Kellert's 1993 (called, I think "In the wake of chaos") book argues that in a sense chaos theory does undermine the assumption of determinism. But I find his argument unconvincing. This paragraph should be left out as any philosophy related to chaos theory is almost certainly original research. The field is a mess, and there is lamentably little written on chaos from a philosophical perspective. 158.143.86.159 (talk) 14:27, 28 July 2009 (UTC)

## "Renewed" physiology

I removed User:Lapaz's addition of the sentence "The emergence of chaos theory renewed physiology in the 1980s." from Physiology. I wasn't aware the physiology needed any renewing in the 1980s. I see a similar sentence here, although there is more detail (the sentence here is "Chaos theory thereafter renewed physiology in the 1980s, for example in the study of pathological cardiac cycles." Is it possible to add a reference for this fact, and maybe change the wording a bit, to avoid implying that physiology was dead? - Enuja (talk) 00:46, 19 January 2008 (UTC)

Formulation may have been poorly chosen, but chaos theory did impact physiology and modify approaches, in particular by boosting mathematical researches. There is a source concerning the eye tracking disorder here in this article. My original source was James Gleick's Chaos: Making a New Science, the chapter at the end on "Internal Rythms". Lapaz (talk) 19:50, 23 January 2008 (UTC)
I suggest you cite the source, then. In this article, how about "Chaos theory provided new computational approaches for physiology, for example in the study of pathological cardiac cycles.[1]"
• ^ Gleick, J. Chaos: Making a New Science 1987.
You can name the ref and easily re-use it for everything that comes from that book. I regularly go look at WP:Footnotes to figure all that out. - Enuja (talk) 20:06, 23 January 2008 (UTC)

## Statistics?

The article contains the sentence:

As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics.

What is the meaning of the term "statistics" here? A measure of the average amount of time the system state is, on the long run, in a Lebesgue-measurable subset of phase space? It seems that this would contradict a statement on pp. 168–169 of Gleick (1988 Penguin paperback edition). Does anyone have a citable source for this statement?  --Lambiam 18:42, 28 March 2008 (UTC)

Could somebody please translate the very first sentences of this article. As soon as I got to 'nonlinear' it was like digging underground in the dark with the earth coming in on you. Fuck, if some of you lads had your way this entire article would be a stream of equations! I just want to know the broader applications of this chaos theory stuff in simple English. Tanks. 86.42.102.87 (talk) 14:09, 18 April 2008 (UTC)

See the information under A section a simpler terms? above. Gandalf61 (talk) 14:28, 18 April 2008 (UTC)
I have taken a look at the first sentence:
• In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect).
Couldn't this be in simple English:
I do think such an introduction would Wikipedia: The first sentences and introduction should be understandable to a larger audience.-- Mdd (talk) 19:58, 18 April 2008 (UTC)
The words "specific kind" would be potentially misleading; it is in general not possible to tell whether a given system is chaotic or has merely very complex, but nevertheless non-chaotic, dynamics.
It is essential that the systems are dynamical. For example, it would be nonsensical to consider the question whether a given coordinate system is chaotic. The word "dynamical" must therefore not be omitted from the first sentence. We might add an explanatory phrase (e.g., "dynamical systems – that is, systems whose state evolves with time –"), although I agree with Gandalf61 that a reader who does not understand the term should either just ignore it, or follow the link. (However, the lede of Dynamical system is, unfortunately, not as accessible as it should be. Until that is fixed, a slightly better article to link to is perhaps Dynamical system (definition).)
The word "nonlinear" is less essential and could be omitted from the lede; in fact, it is more informative to state, somewhere in the body of the article, the theorem that linear dynamic systems are not chaotic, which implies that chaotic systems are nonlinear. There are more theorems that could be mentioned, such as (if I'm not mistaken) that the phase space has to have dimension ≥ 3 for chaos to arise. A natural place for this is the section Chaotic dynamics.
In my opinion, the words "in physics" could also be struck, just as we also do not state: "In mathematics and physics, spectral theory is ...", even though spectral theory has applications in physics, as in the models of the hydrogen atom. The theory is mathematical, and as far as chaos theory relates to physics, it is actually to mathematical models embodying theories of physics. The apparent chaotic behaviour of various natural systems may be explained by a chaos-theoretical analysis of such mathematical models.  --Lambiam 07:17, 25 April 2008 (UTC)
I agree with Lambian's points. Minor clarification - a continuous dynamical system on a plane space must have a phase space with at least 3 dimensions to exhibit chaotic behaviour - this is a consequence of the Poincaré–Bendixson theorem. However, a discrete dynamical system (such as the logistic map) or a continuous dynamical system on, for example, a torus can exhibit chaotic behaviour in a phase space of 1 or 2 dimensions. Gandalf61 (talk) 08:25, 25 April 2008 (UTC)
Thanks, my proposal wasn't such a good idea. I only just noticed that there actually is a Chaos theory article in simple English. This makes a difference to me, because this gives the opportunity to make a difference in a simple and more complicated description.
Now I do agree with Lambian, that we should loose the "In mathematics and physics" frase in both descriptions. The introduction should state that chaos theory "has mayor applications in physics", and that "the theory is mathematical" or as Lambian states "it is to mathematical models embodying theories of physics". -- Mdd (talk) 11:51, 25 April 2008 (UTC)

## Dasavathaaram

The first question is copied from User talk:Mdd

The film's theme is based upon the chaos theory. Why do you insist on removing it? Universal Hero (talk) 19:46, 15 June 2008 (UTC)

There are good reasons why this movie is removed from the list already some seven times here: There are tons of movies about chaos theory. At least a dozend movies have been removed there before the last year. Now there is one new film Dasavathaaram, which is a Tamil language feature film...!? What does that mean...? Certainly not that we should mention it all at once. No. The listing of movies here is just selection of the most important movies ever made. The Dasavathaaram movie certainly isn't (yet). I will make you a deal. If he receives an Oscar next year, we will add the movie to the list. Until then I consider it just linkspam to get any attention to this what ever Tamil language feature film...!? -- Marcel Douwe Dekker (talk) 20:21, 15 June 2008 (UTC)

Double standards- Because Jurasic park is an American dream, it gets a position in Wikipedia!! Dasavatharam is more apt to be there.Wikipedia is a dynamic knowledge portal and has to be nutural to reality. Sad commentary that it ignores Dasavaatharam.Ramesh

It is rather foolish of you to dismisss a Tamil film as you have done. Dasvathaaram is the most expensive Indian film ever to be produced (inc. Bollywood etc) and features the lead actor in ten roles, breaking the world record for the highest amount of roles in a feature film. Furthermore the film is being distributed by Walt Disney and Sony, becoming the biggest film in Indian cinema ever. the film is the turning point in the non-Western belief of the chaos theory, becoming the first film to explore this issue in the non-Western world. Universal Hero (talk) 23:07, 15 June 2008 (UTC)
Ok, why the rush. We look at it next year. The Wikipedia is not to promote all greater things to come. -- Marcel Douwe Dekker (talk) 23:16, 15 June 2008 (UTC)
When the people who have watched and understood the plot and could genuinely relate it to the Chaos theory, why would you not listen to the so many credible people are vouching for it? Oscars alone don't rule the movie world. Any movie - good or bad, significant or not, whether you know the language or not, if it has to have a place on the WIki, it has to be there. Since you have not watched Dasa, it does not become any less important. A movie a MOVIE wherever it is made and Oscar or Hollywood alone don't hold the bastion. Dasavatharam deserves a place is a loud call. If you are refuting, go watch the movie. 207.229.184.110 (talk) 02:38, 23 June 2008 (UTC)
It doesn't matter whether anyone is refuting it; it's just that you can't make wikipedia content out of WP:OR; what's required is WP:V, WP:RS. Re-read those and let us know what you think. Dicklyon (talk) 03:24, 23 June 2008 (UTC)
In what way exactly is the film based upon chaos theory ? Reading the plot synopsis in our Dasavathaaram article, I can see no connection at all. Do you have a reliable source that discussses the film's connection with chaos theory ? Gandalf61 (talk) 21:22, 15 June 2008 (UTC)
The plot section is still under active construction. If you see the film, the element of chaos theory is mentioned by the characters several times, as well as mentioning the butterfly effect. Furthermore, why don't you opt for a Google search on "dasavathaaram chaos theory" - I'm sure that'll give you the needed info. Universal Hero (talk) 23:07, 15 June 2008 (UTC)
Even if this isn't a googlebomb, it's likely that none of those pages is both reliable and actually says the movie is related to chaos theory. It's the job of the editor adding information to support it, not the job of reviewing editors to find support for factoids. — Arthur Rubin (talk) 01:23, 16 June 2008 (UTC)
Several attempts have been made to add this film to The Butterfly Effect article as well. The lack of sources, the inability to explain just what the relevance of the film is (I notice that theme has turned into mentioned several times above), and the repeated attempts to add without bothering to discuss make this look more and more like spam to me. Gandalf61 (talk) 06:48, 17 June 2008 (UTC)
What is the chaos in adding Dasavathaaram or any other film that relates to chaos theory into the article? Adding Dasavatharam to the list isn't going to be touted as promoting the film nor is there a special rule on adding films that win the Oscars to be mentioned in articles. A film is a film, whether good or bad. Dasavathaaram perfectly portrays the Chaos Theory and Butterfly Effect in a well organized manner to the locals of India in the three prominent local languages of the country: Tamil, Hindi, and Telugu. I highly doubt those films that are mentioned in the article already would be known to Wikipedia users in those regional areas. Wikipedia is used by everyone all over the world, knowing its part of the World Wide Web and we must keep it usable for everyone. Dasavathaaram doesn't need the Oscars to prove it an honourable mention. If you search videos of Dasavathaaram Soundtrack Audio Launch you can see yourself how even Jackie Chan speaks about the film in the function. Bottom line, adding Dasavatharam to the list is NOT spam or vandalism, it is only the implementation of the article itself and of course, Wikipedia. It is something that exists in the world and must be mentioned, and the only encyclopedia ever to explain the most stuff in the world is Wikipedia. What some of you are doing is just hogging the article to yourself which is pretty selfish. Respect the willingness of other Wikipedians to add viable and reasonable information and do not prevent them from doing so. Eelam Stylez (talk) 03:06, 21 June 2008 (UTC)

Dasavathaaram is a movie whose plot is STRICTLY based on chaos theory. As you see one of its initial scenes starts with a butterfly symbolic to the butterfly effect. The Scriptwriter explains what is the theory and his story is based on this theory, and i see it's much more relevant than theButterfly Effect and its stupid that you consider it as an ordinary regional tamil movie. Why Indians are treated so? So can you only approve a hollywood movie? That is RIDICULOUS! Harikrishnan (talk) 03:06, 21 June 2008 (UTC)?

So provide a source. That is all that is required. Provide just one independent reliable source that discusses the importance of chaos theory as a theme of this film, and then there should be no objections to its inclusion in the article. Gandalf61 (talk) 16:13, 21 June 2008 (UTC)

This movie has a much bigger profile than any of the hollywood movies listed as examples of the "Chaos theory" in popular media. The opposition of these American editors is based on little other than bigotry against anything that is Indian.There is no other explanation --59.189.44.116 (talk) 06:00, 25 June 2008 (UTC)

So supply a source, even a Tamil language source. It hasn't been done, yet. — Arthur Rubin (talk) 06:52, 25 June 2008 (UTC)

This edit shows the attitude of the anti-Indian bigots who control this page. I added a source for Dasavatharam showing how it relates to the Chaos Theory and added fact tags to the other movies, yet User:Arthur Rubin reverted the edits without so much as an explanation. Funnily enough, these people who are foaming at the mouth over the inclusion of Dasavatharam don't require sources for the American/English langauge movies and books. It speaks for itself--59.189.44.116 (talk) 07:09, 25 June 2008 (UTC)

The movie has been added and removed about 50 times in the past two weeks. -- Marcel Douwe Dekker (talk) 09:32, 25 June 2008 (UTC)
The other movies mention Chaos theory in their article, and the information is sourced there. Your article had mentioned it, but without sources, even Tamil language sources. — Arthur Rubin (talk) 12:20, 25 June 2008 (UTC)

I'm a bit baffled by this. 59.189.44.116 is doing himself no favours by ranting; but whats all this about asking for RS to demonstrate that films are indeed about the effect? I've just perused the butterfly effect film page; I see no RS there saying its about BE. This smells of double standards. If the objection is that we've got enough films about BE and Das is non-notable then lets stick to that argument, and not invent implausible new arguments William M. Connolley (talk) 22:36, 25 June 2008 (UTC)

And I've just looked at The Science of Sleep which is liked from Chaos theory. WTF!?! Why? It doesn't even claim any connection William M. Connolley (talk) 22:39, 25 June 2008 (UTC)

That only means that it too, may be challenged and removed if no reliable source can be found. Every film that has a central plot revolved around World War II is not listed on that page, nor articles for cowboy, murder or love, much more popular themes. Finding one or two reliable source to back a claim is not too much to ask. Requests to establish some notability are made for a purpose, to verify to other editors the significance of a link, in this case, between an unreleased, Indian film and a mathematical theory. My preferences would be to only list documentaries that are about chaos theory. - Shiftchange (talk) 23:19, 25 June 2008 (UTC)

Well I've tested what seems to me a rather odd idea, by rm'ing all films that don't have RS to link them to chaos. Thats leaves only one. I was tempted to do the books too, but that would have meant taking out Gleick, which would be veeeeerry odd. I'll leave that to the people who actually believe what they've written above William M. Connolley (talk) 18:11, 26 June 2008 (UTC)

"The butterfly effect" is not exactly related to chaos theory, only to the assertion that small actions can lead to large unpredictable effects. — Arthur Rubin (talk) 18:33, 26 June 2008 (UTC)
So, are you going to take out Gleicks book? William M. Connolley (talk) 18:42, 26 June 2008 (UTC)
Maybe the best way to integrate wikilinks to related films is either in a Cultural references section, or better yet in a template at the bottom of the page. A good example would be Template:Peak oil listing films, books and people related to peak oil. Having a template for chaos theory would also clear the large see also list on this page. - Shiftchange (talk) 05:12, 27 June 2008 (UTC)
The template/Cultural References section sounds good. I request the Wikipedians who maintain this article consider that to clear the controversy over Dasavathaaram. Eelam Stylez (talk) 3:11, 30 June 2008 (UTC)

## Hash functions

Maybe the arcticle should mention hash functions and the avalanche effect, as somewhat related topics. Just an idea. --Azazell0 (talk) 19:49, 9 July 2008 (UTC)

## why does nonlinear dynamics redirect here?

All chaotic systems are nonlinear, but not all nonlinear systems are chaotic. Only non-linear systems with positive feedback are chaotic (i.e. they must have a positive Lyapunov exponent). Systems that do not have feedback are linear systems. Systems that have feedback are non-linear systems. Systems that do not have any positive lyapunov exponents (i.e. have only negative feedback) are categorically NOT chaotic (per definition in chaos theory). Chaos theory only deals with systems that are chaotic (per definition in chaos theory). Therefore, re-directing nonlinear systems to chaos theory is categorically wrong. And I presume that, by the same mistake, much of this article's content has nothing to do with chaos theory proper (i.e. should be in the (non-existent) article on nonlinear dynamics).

To put it another way, systems that have feedback but do not have any positive lyapunov exponents -- such as, oh say, the planets orbiting around the sun, lasers, predator-prey relationships, and the microprocessor inside the very computer that i'm typing this on -- are categorically NOT linear dynamical systems and categorically NOT chaotic dynamical systems. So where do they go? Well I can tell you two places that they clearly DON'T go:

• an article on linear dynamics
• an article on chaotic dynamics

Kevin Baastalk 17:06, 18 July 2008 (UTC)

Non-linear dynamical system redirects to dynamical system. In the absence of an actual article on non-linear dynamics (as distinct from chaos theory), I am changing non-linear dynamics to redirect to dynamical system as well. Gandalf61 (talk) 17:59, 18 July 2008 (UTC)

Reference #35 is a broken link. —Preceding unsigned comment added by 70.168.46.253 (talk) 14:25, 26 August 2008 (UTC)

## Deleting page "Chaos (physics)"

I was planning on merging Chaos (physics) into this page. Having read the Chaos (physics) article in more detail, I can't see anything in it worth keeping that isn't duplication of the Chaos theory page - however, I don't know enough about the subject to really be sure. The text of the Chaos (physics) article is below in case anyone wants to add anything from it into this article.

Chaos in physics is often considered analogous to thermodynamic entropy. Chaos is a poetic or metaphysical concept evoking a sense of discord, whereas entropy is a concretely defined function of a physical system. See entropy for the mathematical quantification of the disorder in a system.
The term "chaos", as commonly used, denotes utter confusion, an incomprehensible and heterogeneous mess. This intuitive notion is at odds with the famous Second Law of Thermodynamics, which states that entropy cannot decrease in a closed system. Maximized entropy always corresponds to apparent homogeneity in a system. Any random disturbance of a homogeneous system results in no meaningful change, therefore scientists will say the randomness, i.e. chaos, is maximized. Such systems are observed as being isotropic.
As with any scientific concept or mathematical abstraction, entropy may not be equally applicable in every situation. For example, it is unknown whether protons may remain forever free and unchanged, or whether they are subject to destruction by cosmological randomness.
Chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. Among the characteristics of chaotic systems, described below, is sensitivity to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the system is deterministic in the sense that it is well defined and contains no random parameters.
However, chaos as defined in physics is strongly contrasted with the common definition of chaos. Chaotic systems, with no central control, are able to create "order"; that is to say, they may form a pattern that humans recognize. Biological systems are well-known examples. Potential applications are found in nanotechnology, where self-assembling systems have been developed.

Thanks! Djr32 (talk) 17:55, 25 October 2008 (UTC)

## Introductory Paragraph

The introductory paragraph is merely a clumsy fanboy exercise in jargon-dropping. It is not an honest attempt to explain a concept to someone unfamiliar with it, in familiar terms. Some half-dozen technical concepts are referred to, but left unexplained and unreferenced [probably just as well as an uninitiated reader would never get to the end this paragraph]. This paragraph is at best redundant. Anyone familiar with these concepts already has some notion of what "chaos theory" is. Those that don’t will be none the wiser, and will probably stop reading. This paragraph does define nor explain one single concept.

Chaos is not confined to dynamical systems, so the reference to that subject in an introductory paragraph is unenlightening and relevant. Surely the "May" in "May exhibit dynamics …", shouldn’t be there, as this sensitivity is a defining feature of so-called chaotic systems. Not all "exponential growth of perturbations in the initial conditions blah blah …" is indicative of chaos, so this statement is both pretentious and irrelevant. The second paragraph is no more enlightening. It doesn’t really say anything, but terms like "chaos-theoretic analysis" sound oh so-good, and I’ll resist the temptation to say "its information-theoretic content is null". Moreover the term "chaos theory" better describes a zeitgeist, now terminal, that produced a lot of flaky pretentious nonsense, and should be avoided, unless trying to impress girls at parties.TitusCarus (talk) 09:51, 13 November 2008 (UTC)

Thanks for the time spent taken to comment, but that was a validictory one. I've deleted my pass-word, so I cannot use my account. Editing Wikipedia articles is not an efficient use of my time. PS. I'm always polite. Mr Lucretius. —Preceding unsigned comment added by 86.27.192.219 (talk) 11:55, 13 November 2008 (UTC)

## "chaotic"

humans call everything that's transcendent to us or out of our observational reach, "chaotic". it's a ridiculous, overused term. 68.46.139.114 (talk) 03:30, 22 November 2008 (UTC)

## Subject of article does not exist

I am dead serious! There is no branch of mathematics (despite what the Mathematics article says) called "Chaos Theory".

The relevant branch of mathematics is Dynamical Systems, in which "chaos" is a concept. Period.

It's not even a well-defined mathematical concept, since there a several closely related but distinct concepts that have been used to convey the idea of chaos.

This article perpetuates the myth that there is such a branch of mathematics. The fact that the phrase "chaos theory" has appeared many times in print (and perhaps has been mentioned several times on the NUMB3RS TV show) is not evidence that there is such a branch of mathematics.

It's fine to have an article that discusses this concept. But it's not fine to perpetuate a myth by making it appear as fact.75.61.110.161 (talk) 17:25, 5 March 2009 (UTC)

As far as I can see, this article doesn't say anywhere that chaos theory is a "branch of mathematics". If you disagree with the contents of the Mathematics article, then you could take this up at Talk:Mathematics. Gandalf61 (talk) 17:40, 5 March 2009 (UTC)
I agree. Chaos theory is not limited to the field of mathematics. Physics and philosophy plays heavily in understanding, cogitating, and making use of the concept. I have changed the lede to reflect this.—αrgumziω ϝ 18:27, 13 November 2009 (UTC)

## Maybe a reference to Jurassic park?

About how Ian Malcolm was obsessed with it? Or perhaps its not needed, just getting my two cents in —Preceding unsigned comment added by 59.92.57.171 (talk) 16:40, 15 March 2009 (UTC)

I must agree I would like to see a "cultural reference" section... maybe this has been discussed in other movies and songs too? 03:58, 7 November 2009 (UTC)~ —Preceding unsigned comment added by 65.74.25.20 (talk)

I'm not wild about it, but I did create such a section, if anyone wants to populate it. -- Radagast3 (talk) 03:40, 7 March 2010 (UTC)

## Definition of chaos theory

As I understand it, the definition of chaotic dynamics given here is contentious. It is a modified version of that found in Devaney's 1989 book Introduction to Chaotic Dynamical Systems. This definition has been criticised by, for example, Peter Smith in his 1998 Explaining Chaos. As far as I am aware there is no such thing as a canonical formal definition of what it means for something to be a chaotic system. This should perhaps be reflected in the tone of that section. Incompetnce (talk) 14:37, 28 July 2009 (UTC)

Agreed. I've reworded slightly to take this into account. However, the article still needs a lot of work, I believe. - Radagast3 (talk) 08:20, 12 October 2009 (UTC)

## Unpredictability

As this diff shows, it is clear somebody has a problem with a particular sentence. I demand that we engage in rational discussion so that the source of the problem can be explicated for all to witness. That isn't too much to ask, now is it? My cards are on the table: "unpredictability" warrants a clear definition in the terms of chaos theory as Charlotte Werndl describes in her paper. What is so "very unclear" about this, hm? I am more than willing to reword the sentence in question to suit the superlative tastes of any who find fault with it (it may need greater elaboration, for instance), but I certainly will not accept that it be removed from the article.—αrgumziω ϝ 17:58, 14 November 2009 (UTC)

If you want a rational discussion, I suggest you drop the bad tempered attitude. I removed your proposed definition because I think it is too complex and opaque to be included in the "Overview" section of the article. I would have no problem if it was included in a later section of the article, with appropriate expansion and explanation, but as a stand-alone definition of "unpredictability" it is far from clear. For example, can you explain what the terms "bundle of initial conditions", "area of interest" and "probabilistically irrelevant" mean in that definition ? Gandalf61 (talk) 18:13, 14 November 2009 (UTC)
I think that as an individual who is endowed with a justifiably vigorous tone due to the salient absence (!) of discussion on this hitherto,—for you did delete the sentence before (linked above) with the amusing edit summary: "shorten and clarify"—I need not fulfill your suggestion to drop whatever this "bad tempered [sic] attitude" is. And no, I am not an example of this, at any rate. Now, with that out of the way, I take your suggestion to heart, and think that I will have to create a new section that specifically deals with unpredictability because it is cardinal to any proper understanding of chaotic systems as currently conceived (and the terms for which you reasonably request for further treatment most definitely would be required for this). Therefore, I will proceed to delete the sentence. I will post a new subsection later, though not necessarily in a finalized form. You have my thanks for finally broaching your concerns rather than sweeping them under the rug.—αrgumziω ϝ 23:31, 14 November 2009 (UTC)
I seem to have wandered into this debate by accident, by re-deleting the confusing sentence.
A section on unpredictability from a philosophical point of view could be good (particularly given the fact that philosophy has been included in the lead para). However, I have some doubts about the reliance on Werndl's article, and her distinction between different kinds of unpredictability seems to me non-standard. I would suggest (1) not using it unless we can find additional support for her approach, and (2) rewording what she says in simple English, if we do use it (but it would be better to use a more standard source). Werndl also uses her own (novel, she says) definition of chaos in terms of mixing, which differs from the definition this article uses (although I note that Vellekoop and Berglund, in the American Mathematical Monthly of 1994, showed that mixing implies chaos in the usual sense on intervals of the reals). -- Radagast3 (talk) 01:33, 15 November 2009 (UTC)
Yes, I agree that relying on Werndl would be a bad thing, leading to distortion, inaccuracy, misinformation, etc. However, she makes it plain that unpredictability is central to understanding chaos, which isn't exactly controversial (though her definition of chaos as mixing is certainly novel); therefore, in my view it requires some elaboration in the article. Not to mention that she isn't the only one to mention unpredictability when chaos raises its horns (and you point this out as well), so perhaps something is to be said for it. I've been quite busy, but I'll see what else I can find regarding unpredictability, so we can get the section started. Any comments, thoughts, additions, criticisms, and so on are very much appreciated, naturally.—αrgumziω ϝ 04:52, 17 November 2009 (UTC)
Well you can use high level esoteric math all you want. The reason it's unpredictable is because at every point on the path you don't have a one-to-one and onto relationship between f(t+dt) and f(t). Doesn't matter how complicated and high-fa luting(sp?) you say it (and i'm sure there are many ways to say it that way), it still boils down to that. And I believe the point of any informative article is to be clear, so let's aim for that. Kevin Baastalk 14:53, 18 November 2009 (UTC)
Actually, that's not the reason. Determinism means that f(t) uniquely defines f(t+dt). So we may need to make the article a little clearer. ---- Radagast3 (talk) 20:16, 18 November 2009 (UTC)
If you have a continuum of points from 0 to 1 mapping to a contiunuum of points from 0 to 2, you can say that x = 2y, so yeah,it's "deterministic" in that sense. but if you use the same measuring stick on each one to tell what point on one continuum maps to what point on the one that's twice as long, even as the size of your measuring stick shrinks to infinitesimal, the smaller continuum still isn't going to give you complete information about the larger one. You're always missing a bit. (And I mean that precisely: exactly one bit.) That's what's happening in a strange attractor, except it's going on continuously. (and both ways: small to large and large to small, just in different dimensions) that's why it's an "attractor", because the set of starting points is larger than the basin of attraction -- not one-to-one and onto. but there's also an expansion going on (short line to long line) as evidenced by the positive Lyapunov exponent. (This provides a constant in-flux of new information. Which is why the weather is unpredictable.) So in one dimension it's not one-to-one and in another it's not onto. Kevin Baastalk 18:03, 19 November 2009 (UTC)
Also, btw, a logical argument of why if the paths are to diverge like they do they MUST do it exponentially on average: because provided the phase space structure is constant through time, the average amount of information it produces or consumes must likewise be constant through time, and a constant information production/consumption rate means exponential divergence. one bit is 2 possibilities, two bits is 4 possibilities, three bits is 8 possibilities, etc. exponential divergence. Just a side note. But it serves to clarify what I'm saying. Kevin Baastalk 18:17, 19 November 2009 (UTC)
And yes, as your dt shrinks to infinitesimal, so does the amount of information you lose/gain. but the point is that's not the same as zero. di/dt <> 0. Kevin Baastalk 19:09, 19 November 2009 (UTC)

Personally I think a philosophical section on "unpredictability" is something I'd be entirely bored with and actually find silly. Unpredictability is no more in need of philosophical clarification than predictability.* They are just two sides of the same coin. periodic attractors are predictable because they consume information and they don't produce any. strange attractors are unpredictable because they consume AND produce information. That's what the lyapunov exponents mean: they are the rates of information consumption/production. I showed that pretty clearly with my measuring stick stuff above. it's not differentially one-to-one. when you have one point that becomes two (differentially speaking), that's ambiguousness. Lack of information about the future.** That's the definition of unpredictability. There's no leap, there's no thought, there's no calculation, it's just the definition. Just as predictability is the line of possibilities shrinking with time. No mystery there. Nothing new there, either. Just a sign flip (negative/positive).
*(thou it scares people more and they end up inveting all sorts of fantasies to cope w/their fear, which they then have a hard time separating from fact)
**(said information doesn't even exist yet; that infinitesimal quantity of info represents the ratio of the (above-mentioned) continuum's length at t vs. t+dt and that span hasn't been traversed so there is no bit yet. just like in predictability the information isn't destroyed/consumed until time progresses))
Kevin Baastalk 19:36, 19 November 2009 (UTC)

Can anyone please tell me the name of the hypothesis where a thing changes, because you looked at it, irregardless of an operational standard of non-interference? Thank you 67.176.91.13 (talk) 07:30, 20 November 2009 (UTC) 2009-11-30 T00:30 MST

Butterfly effect ? Uncertainty principle ? Wave function collapse ? See Copenhagen interpretation and Schrödinger's cat. Gandalf61 (talk) 09:30, 20 November 2009 (UTC)
It sounds like the Wave function collapse / Copenhagen interpretation and Schrödinger's cat. I don't see how the "wave function collapse." is so profound. The wave function is a frickin' probability manifold. It's like saying, "before I roll the die, it has an equal chance of ending on 1,2,3,4,5, or 6. After I roll it, it will take on exactly one of those values." which is just the definition of probability. Observing is the same as rolling because it's not passive - the eye and everything else is a machine w/moving parts and you're connecting those moving parts up to the thing being observed so in effect by observing you're rolling the die. It's really a physical necessity and there's nothing special about it. Schorndiger's cat, IMO, is just a fanciful - and superfluous - explanation brought about by a failure to understand this. Kevin Baastalk 14:56, 20 November 2009 (UTC)

## Condition #3

I've deleted the unsourced modification that suggested that condition #3 in the Devaney definition of chaos was redundant. For evidence, see the numerous books on chaos that use this definition, or the cited reference by Medio and Lines (which supports the current wording with "weaker"). The editor making the change seems to be confused by the fact that #2 and #3 imply #1 (i.e. #1 is redundant). -- Radagast3 (talk) 05:09, 25 November 2009 (UTC)

## its periodic orbits must be dense (again)

This isn't very clear. Does it mean that any point in phase space is arbitrarily close to an orbit of finite period? I suppose that the period would have to go infinite as the arbitrary closeness was reduced to zero in order to exhibit aperiodic behaviour, and the that periodic orbits would all have to be repellers - is that right???

Also, what's the significance of this condition? And why do only some authors specify it? --catslash (talk) 19:57, 15 December 2009 (UTC)

We could explain this better. The significance comes from the fact that conditions #2 and #3 imply #1. -- Radagast3 (talk) 22:24, 15 December 2009 (UTC)

Thanks, that's a little bit clearer. --catslash (talk) 00:53, 16 December 2009 (UTC)

I've tried to clarify a little more. There does seem to be a shortage of good books to refer to that are more technical than, say, Gleick, but less technical than, say, Devaney. And there are still paragraphs in the article that could be a lot clearer. -- Radagast3 (talk) 23:24, 18 December 2009 (UTC)
I hope it's clearer now -- I've reorganised to clarify and avoid repetition. -- Radagast3 (talk) 11:08, 19 December 2009 (UTC)

The text says: For example, an irrational rotation of the circle is topologically transitive, but does not have dense periodic orbits, and hence does not have sensitive dependence on initial conditions. This statement could be improved in stating: "For example, the open set of irrational rotations of the circle...", since it is not clear, that the rational rotations are excluded, and the rational rotations of the circle correspond to periodic orbits which are dense and will approach all irrational orbits arbitrarily closely. Sigi E (talk) 12:45, 19 December 2010 (UTC)

## Alan Turing

I'm shocked to see no mention of Alan Turing in this article !! —Preceding unsigned comment added by 217.33.199.76 (talk) 13:48, 19 January 2010 (UTC)
Why ? Which particular aspect of Turing's work do you think should be mentioned in this article ? Gandalf61 (talk) 14:11, 19 January 2010 (UTC)
I think he's referring to Turing's work on the chemical basis of morphogenesis. (It uses a @^#load of complicated diff eq.) though i wouldn't know the answer to your question, specifically. Kevin Baastalk 22:08, 3 February 2010 (UTC)

## Terminology Question

Is a nonlinear dynamical system still considered 'chaotic' over time ranges where it has not bifurcated or where it has not become unpredictable. Similarly for a dynamical system whose chaotic behavior depends on its parameters, are only those systems that have chaotic parameters be referred to as 'chaotic' or is the general class of the dynamical system 'chaotic'. OR is 'chaos' a term that should not describe systems adn only refer to regions where a dynamical systems behavior is chaotic? For example: I select parameter ranges that are on the 'edge of chaos' for a system or I study nonchaotic regions of a function? Mrdthree (talk) 05:34, 3 February 2010 (UTC)

to answer the first question: the classification of chaotic or non-chaotic is over all possible time ranges - i.e. if the system of diff. equations will ever be chaotic then it is always chaotic. and vice-versa; it is either always or never chaotic (for the given parameterization). you can classify a system as chaotic or non-chaotic on the basis of it's lyapunov exponents. (in particular i believe it must have at least 1 positive and 1 negative lyapunov exponent) And lyapunov exponents do not depend on what value or range you "put in for t".
(without a positive lyapunov exponent, there would be no instability ("unstable manifolds") and thus it clearly would not be chaotic, and without a negative lyapunov exponent, it could be said to be merely "random" rather than chaotic, as there would be no basin of attraction or anything of the sort; no stable manifolds)
the general class of dynamical systems is not chaotic. only those parameters, or that "mode", if you will, wherein the system behaves "chaoticly" is said to be chaotic.
so then i suppose to your next question, "chaos" only refers to regions of parameter space wherein a system of dif. eq. is chaotic. (e.g. according to lyapunov exponents) (and btw, i'm using "system of dif. eq" to mean "dynamical system" just to be extra clear/careful w/my wording.)
so i suppose in sum when and only when the lyapunov exponents meet certain conditions is a dynamical system said to be "chaotic".
I hope this answers your question and any one please correct me if I made any mistakes. Kevin Baastalk 21:51, 3 February 2010 (UTC)
I think that was neccessary but not sufficient. there also has to be "mixing" of stable and unstable manifolds for there to be chaos. (you can't have it just diverge in one dimension and converge in another. that would just be x=random, y = 0.) and for that you have to at least be able to "twist" or "bend" in vector space, which would mean that you need at least three dimensions. (ignoring fractals) Kevin Baastalk 22:49, 3 February 2010 (UTC)
That works. The discussion I was having related to a computer simulation of a nonlinear system which you run for say 1000 timesteps. Depending on the parameterization of the system it may or may not be well behaved for those 1000 timesteps (say easily approximated by a linear system), but for all parameters and time scales outside of this range, the system is understood to behave 'chaotically' (perhaps except a a few finite/trivial parameterizations). So my take away is that if the particular parameterization of a dynamical system is anywhere chaotic, it is a chaotic system. Thanks. Although I guess he could argue he was only interested in a function with a restricted domain. Mrdthree (talk) 03:42, 11 February 2010 (UTC)

## Dubious

I take issue with the sentence: "This happens even though these systems are deterministic, meaning that their future dynamics are fully determined by their initial conditions, with no random elements involved." esp. that:

1. it is unsupported by any citation, or any material in the body of the article
2. i believe it is simply wrong (on multiple counts)

I have added tags respective to this. and i will clarify point 2 momentarily. Kevin Baastalk 21:41, 10 March 2010 (UTC)

taken literally and mathematically, determinism as defined in the above sentence implies perfect predictability, and imperfect predictability precludes determinism, in addition to implying randomness.

from my understanding "deterministic" in chaotic systems refers to the fact that the system is defined by a set of finite-length diff. eqs. which do not have a stochastic term. this does not imply determinism in the sense used in the sentence i take issue with (perfect causal determinancy). nor does the absence of a stochastic term neccessarily imply the absence of "randomness".

in fact, ergodic theory, among other mathematical principles, shows that in a chaotic system, the mutual information between the state at time t and the state at time t+dt is not equal to the total information at time t or time t+dt (or put otherwise, the joint entropy is greater than either of the individual entropies). which means that, statistically speaking, as far as time t is concerned there is an element of "randomness" in the state at time t+dt and vice versa. which is the same thing as saying one is not wholly predictable from the other. in fact these things are all different ways to say what is mathematically the same thing. a thing can not be one and not the other (or not one, yet another).

this also means that their future states are not fully determined by their initial states (or "conditions"). though granted, the sentence did not actually say that -- it said future "dynamics". and the sentence is actually wrong in this regard, too (thou only half so): while the future dynamics are fully determined, they are not determined by the initial conditions (initial values of the variables), the dynamics are determined by the equations. Kevin Baastalk 21:53, 10 March 2010 (UTC)

Oh, and i consider this pretty important, as to me it amounts to a big misunderstanding in the lead that promulgates a wrong (and simply illogical) misconception of an important nuance of chaos theory. Kevin Baastalk 22:08, 10 March 2010 (UTC)
I've added a ref. There are hundreds of others that all say the same thing: "deterministic but not predictable." -- Radagast3 (talk) 22:13, 10 March 2010 (UTC)
If the problems i mentioned aren't fixed, then i assure you those refs don't mathematically support what the sentence says. Kevin Baastalk 14:37, 11 March 2010 (UTC)
Indeed. "Deterministic" means deterministic in the mathematical sense i.e. not stohcastic - I have fixed the link so it points to deterministic system (mathematics) instead of deterministic system (philosophy). I have also clarified that the future evolution is determined by the equations of the system as well as the initial state - this seemed obvious to me, but it is possibly a source of confusion. And "not predictable" means that an arbitrarily small perturbation of the initial state leads to significantly different future behaviour, a propery that is encapsulated in the phrase "sensitive to initial conditions". Gandalf61 (talk) 11:52, 11 March 2010 (UTC)
but it is still wrong: the dynamics of the system are not affected by the initial state. Kevin Baastalk 14:37, 11 March 2010 (UTC)
We are obviously using different meanings of the word "dynamics." I have substituted "behaviour," which is less ambiguous. -- Radagast3 (talk) 15:01, 11 March 2010 (UTC)
ya, i think we were. when i read "dynamics" i read the greek "dynamis", meaning change and more precisely affecting and/or being affected by, by extension dynamics to me is how a system changes and responds to change, which is described by something like a phase space representation of the vector field and in any case something that is atemporal (so "future" dynamics is meaningless, unless you are saying that you intend on changing the parameters of the system). i felt like the wording was conflating things and thus introducing confusion/risk of contradiction. I like "behavior" better. really i presume you mean "trajectory" or "state", but then that contradicts the next sentence which says its not predictable. so see when you clarify the terms it seems to reveal contradiction. Kevin Baastalk 15:52, 11 March 2010 (UTC)
Thanks. I think it reads better now.
I tried to get one of the local physicists to explain ergodic theory to me, but I'm not sure I got it. As far as I can tell, though, it formalises the "unpredictability" concept.
Consider the function on [0,1] that maps x to the fractional part of 10x, e.g. 0.14159265 → 0.4159265. Given a given real number x (i.e. infinitely many decimal places), the trajectory is full determined. However, if all we know is that the number starts with 0.141592, then after 6 steps we can no longer say anything at all: the trajectory will look random, but only because we started with finite information about the initial x. -- Radagast3 (talk) 12:26, 11 March 2010 (UTC)
"starting" is irrelevant. it doesn't compare f(0) to f(t), it compares f(t) to f(t+dt). also "finite" isn't relevant either. the put is, like i said, there is ALWAYS information in f(t+dt) that simply is not in f(t). finite precision or not. it doesn't matter if you're using integers, rational number, real numbers, or even complex numbers. (and really you're always using complex numbers because when your working with an infinitesimal (dt), your dx's are all going to be infinitesimals so you're invariably working with infinite precision) there is still information in f(t+dt) that is not in f(t). and when A contains information that is not in B, from the point of view of B that information is the very definition of random. you can start with infinite amount of information of the initial conditions, but at time dt the system will still always have exactly x more, as defined by the Lyapunov exponents. Kevin Baastalk 14:27, 11 March 2010 (UTC)
And actually finite precision systems cannot be chaotic. they can only be periodic, though they may resemble a chaotic system (e.g. lorentz' computer simulation of the lorentz attractor - the simulation is actually periodic - though over a VERY long time scale.) Kevin Baastalk 14:43, 11 March 2010 (UTC)
Actually, finiteness (of information) is the key issue. In my example, if if all we know is that the number starts with 0.141592, then 0.141592 → 0.415921 → 0.159211 → 0.592111 → 0.921111 → 0.211111 → 0.111111 is one possible trajectory, and 0.141592 → 0.415929 → 0.159298 → 0.592987 → 0.929876 → 0.298765 → 0.987654 is another. In this case, each step loses one decimal place worth of the original information, and adds, as you correctly point out, one decimal place worth of new information. After 6 steps, all the original information has been lost, and we are seeing what looks like a random digit generator.
However, if we start knowing infinitely many decimal places -- if we know we have exactly pi–3, then 0.141592... → 0.415926... → 0.159265... → 0.592653... → 0.926535... → 0.265358... → 0.653589... and we have the same amount of information (infinitely many decimal places) at each step (you can't have "x more" than infinity, because infinity + 1 = infinity). Furthermore, the trajectory is totally predictable: at the kth step we just have the decimal expansion of pi, starting at the (k+1)st digit.
Of course, in practice we only ever know finitely many decimal places. -- Radagast3 (talk) 14:52, 11 March 2010 (UTC)
re: infinity + 1 = infinity: yes, but technically infinity <> infinity. since it is undefined and undefined <> undefined. however, if x = x, then x+1 > x. regardless of what x is. by the definition of the set of numbers (namely, "for every number in the set S there is a number exactly 1 greater than it in the set S"). i suppose you might win on the technicality that in this case x <> x. (and might not technically be in the set S, and in any case is not in the set of computable numbers) but my measure theory argument below i think gets more at what i'm saying. Kevin Baastalk 16:10, 11 March 2010 (UTC)
I think you'e wrong on this. To be more specific, the number of digits of information in the digits of a real number is aleph null, which is not undefined, and has 1 + aleph null = aleph null (see cardinal arithmetic).
I'm a computer programmer and to maintain logical consistency in a program NaN <> NaN and if 1+a=a then 1=a-a, which can only be true if a<>a. Kevin Baastalk 16:23, 12 March 2010 (UTC)
Perhaps i'm coming at this from a different perspective. I'm coming at this from the perspective of any other system interacting with the system. From that standpoint the other system always has to measure the system. and one can argue that since a system does not exist except in relation to another (and said other system is really only seeing a "projection" of the system), that is the only physically valid standpoint. and from that standpoint a chaotic system is always producing new information (and destroying old). Kevin Baastalk 15:45, 11 March 2010 (UTC)
I must admit I can't make much sense of that, but it is certainly not a standard perspective or approach in dynamical systems theory. Gandalf61 (talk) 15:51, 11 March 2010 (UTC)
Well I presume that's what is meant when one says that the system isn't fully predictable. (the producing new information part, destroying old info is the predictable part). Kevin Baastalk 15:57, 11 March 2010 (UTC)
You're right there, I think: producing information is very closely linked to unpredictability. However, your use of measure theory seems different from that of researchers in ergodic theory.
I think one source of misunderstanding here is that you seem to be a physicist, and your focus seems to be on dealing with the (unpredictable) system in practice. I'm a pure mathematician, and my focus is on the fact that the system is deterministic in principle. As a pure mathematician, I'm happy to discuss "infinite information," even though I don't have an infinitely large disk to store it on. The key counterintuitive fact about chaos theory, as Gandalf61 rightly said, is that both views are correct. -- Radagast3 (talk) 23:09, 11 March 2010 (UTC)
It's questionable whether this is such thing as "infinite information" and in any case i don't think it applies here. one really only concerns oneself with differences (e.g. kl-divergence), and finite lyapunov exponents means finite information growth (/loss), so in this situation, at least, one is never dealing with infinite information. Kevin Baastalk 16:23, 12 March 2010 (UTC)
In any case it seems like we're saying that its predictable yet not predictable. i think we need to be more clear so that it reads more like "from a certain perspective ("intrinsicly"?) its predictable, from another ("interactively"?) it is not". Kevin Baastalk 16:26, 12 March 2010 (UTC)
And i am kind of coming at this physically rather than pure mathematics, but to me it's more like "in reality..." rather than "in physics...", and i'd argue: "show me an intrinsic representation of a system and i'll show you a bunch of scribbles on a board that are supposed to mean something to me." Kevin Baastalk 16:31, 12 March 2010 (UTC)
Well, I think the standard terminology in the literature of "deterministic but not predictable" captures the two points of view quite well. I'd rather not invent new terminology for this article. -- Radagast3 (talk)
Well I think those minor changes have improved the wording significantly and addressed my concerns. I'm happy with it now. Thanks. Kevin Baastalk 21:35, 12 March 2010 (UTC)

## Asymptotic vs. Transient chaos

I'd like to see a section on this. I don't know any of the formal mathematics behind these phrases; I use them more intuitively in physical systems. I might put in the conceptual/qualitative part myself if no one gets around to it and my main projects get boring. Xurtio (talk) 10:12, 10 August 2010 (UTC)

## Regarding Steven Wolfram on Sensitivity to initial conditions

I have heard Stevens point of view regarding this subject and he appears to completely disagree with the claim of this article that rule 30 is sensitive to the initial conditions. Sensitivity to the initial conditions implies that the complexity evolving in the system depends on the initial input data, in Wolframs 1D cellular automata experiments he starts with the simplest input possible, a single cell. It is from this single cell that infinite complexity self generates within the system, making cellular automata capable of inducing its own complexity and in no way does it require some sort special input to generate it.

Many cellular automata lattice gas fluid dynamic experiments will also demonstrate that the emergent turbulent chaos in the system is also independent to the initial conditions, that despite consistently random (or information rich) initial conditions the same emergent conditions evolve. Making the whole notion of a butterfly effect utter nonsense ! turbulence caused by a butterfly would be averaged to zero in a Wolfram cellular automata lattice gas simulating fluid as it would also be in the real world.

Something to think about next time we are wondering why the butterflies haven't already destroyed the entire universe.

The Zen Monkey.

9:39, 01 Novenmber 2010 (UTC)

Although some cellular automata generate random-looking patterns, they are not examples of mathematical chaos, which arise as described in the article from multidimensional and/or nonlinear dynamical systems. Just because two things look similar does not mean they are examples of the same principle. Rule 30 is irrelevant to this topic until and unless someone can generate the same pattern using a dynamical system. David Spector (talk) 00:24, 17 December 2010 (UTC)
Although the energy of a butterfly's flutter is in principle enough to change the initial sensitive boundary condition of a natural dynamical system, there are several problems with reasoning based on this idea: in practice the energy of such a flutter is too weak to dominate over artefactual 'noise' and system hysteresis; also, in practice the flutter is not applied to a initial (boundary) point but rather to an internal (nonsensitive) point in the system's orbit. In general, there are far more numerous (and far more energetic) inputs to natural dynamical systems than mere flutters of butterfly wings. David Spector (talk) 00:24, 17 December 2010 (UTC)
Your argument that says "the energy of such a flutter is too weak to dominate over artefactual 'noise'" is erroneous. You have no idea what the fluttering might entail. Who knows, Hitler's father might have followed a butterfly when he was young thus leading to a future were he was born. What if that butterfly went left rather than right just enough for Hitler daddy to not notice it. The thing is if you go far enough back in time a butterfly flutter will affect everything. If you moved a proton a few nanomeaters at the conception of our universe humanity would most likely not exist. Random Person 09:33, 2 December 2011 (ECT) — Preceding unsigned comment added by 84.202.66.187 (talk)

## Opening line listing related fields

"Chaos theory is a field of study in mathematics, physics, economics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions." Why not list every field under the sun? Surely Chaos theory applies to meteorology, geology, palaeontology, evolutionary biology, medical science, astrophysics... Does the list need to go on? This: "... field of study in mathematics, physics, economics, and philosophy studying the..." seems to imply that the list is exhaustive, somehow. It is better to just say it is part of mathematics (after all differential/difference equations are part of applied maths, and nonlinear equations produce chaos). We can then add that it has applications to the science. The reason chaos arises in so many different fields of science has to do with the fact that there are systems in science whose underlying behaviours are governed by nonlinear equations. Point-of-fact, it is the mathematics that is the common theme here. Hence that is precisely what Chaos theory is. It may have some origins in Physics but it is fundamentally a mathematical theory: if not pure, then at least applied. I'm going to go ahead and make the change. If someone wants to dispute it, let's discuss it here. Rlinfinity (talk) 03:08, 11 December 2010 (UTC)

## Missing: chaos settling

The article fails to mention the settling or calming of a chaotic dynamical system by the imposition or addition of a periodic or aperiodic external signal, as discussed in the literature.

Examples include patterns of forces applied to systems of coupled pendula or to fluid flows which are capable of moving the systems from chaotic domains to laminar or similar non-chaotic domains.

Such mechanisms probably have important applications in a wide variety of disciplines, including physics and biology. David Spector (talk) 23:30, 16 December 2010 (UTC)

## Some definitions and clarifications for a dumbass (me).

I feel that there are insufficient SIMPLE explanations here and several contradictions in several areas for a simple person (me). Putting them in bullet form.

• What is the relationship, if any, between "random" and "chaotic" systems? For example, a backgammon game depends in part upon the throw of the dice. But does that make it necessarily "random"? And (see below), is it chaotic, both random and chaotic, or neither?
• This sentence to me is completely self-contradictory: "the deterministic nature of these systems does not make them predictable". To me, deterministic systems are those where two systems with identical starting conditions will proceed along identical paths. Does this not make then make them predictable by definition? Is the author saying that knowing that f(x) = y will not help at all determining (predicting) the value of f(x + delta_x) because it is not close to y (does not equal y + delta_y)?
• How would you describe a system that does not obey all three criteria? For example, two games of backgammon that differ in only one throw of the dice, using a reasonably intelligent system that plays the "best" statistical moves, will almost always make those two games diverge wildly from that point. (for example, games with throws a b c X e f g versus a b c Y e f g). So that obeys "sensitivity to initial conditions". But a backgammon game by nature moves in one direction so it won't have "orbits". Or is this entire example irrelevant, and how should I have determined that?
• Another sentence that to me seems incorrect: "The requirement for sensitive dependence on initial conditions implies that there is a set of initial conditions of positive measure which do not converge to a cycle of any length." Huh? I can envisage a poor pseudorandom generator that might converge to a cycle if I started with 12345; and converge to a completely different cycle if I fed it say 12354. So it would be very sensitive to initial conditions yet still "converges to a cycle".
• If you have a system defined by A->B; B->A then it creates A B A B A B ad nauseum, so how can that system be "unstable", even for the \tfrac{5-\sqrt{5}}{8} → \tfrac{5+\sqrt{5}}{8} → \tfrac{5-\sqrt{5}}{8} example given?
• Incidentally, that above example seems numerically wrong in both cases. According to my (Windows) calculator:

1 - 0.90450849718747371205114670859141 = 0.09549150281252628794885329140859 * 4 = 0.38196601125010515179541316563436

and

1 - 0.34549150281252628794885329140859 = 0.65450849718747371205114670859141 * 4 = 2.6180339887498948482045868343656

• Another apparently wrong example: "Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere". Huh? Simple algebra shows that x = 0.8 maps exactly to itself ad infinitum. How is that "chaotic everywhere"?

Old_Wombat (talk) 09:21, 11 May 2011 (UTC)

Let me try to clear up some of your questions by making a few general points:
• In everyday usage, "random" and "chaotic" have more or less the same meaning. But when the term "chaotic" is used in the context of chaos theory, it has a very specific meaning. and refers to deterministic dynamical systems with certain well-defined properties, as listed in the article. The surprising thing is that such systems can be, for practical purposes, unpredictable even thought they are deterministic.
• The dynamical systems studied in chaos theory have a phase space that is continuous and contains an infinite number of possible states. Backgammon and pseudorandom number generators are not good models to use when thinking about chaos theory because they only have a finite number of possible states. A deterministic system that has a finite number of states will, as you say, eventually enter a repeating orbit or reach a steady state, no matter wher it starts from, so it cannot be "chaotic" in the chaos theory sense of the word.
• In the map described as x → 4 x (1 – x), each x is a variable - there are no multiplication signs - so it is
${\displaystyle x\rightarrow 4x-4x^{2}}$
I think you have misunderstood this as x → 4 – 4x, which is, as you say, *not* a chaotic map. Gandalf61 (talk) 10:59, 11 May 2011 (UTC)
As for your second point, in a physical system you can never know x exactly, because of measurement uncertainty, you can only know that it's between (say) x − ∆x and x + ∆x, and if f(x − ∆x) and f(x + ∆x) are very different that's no use in estimating f(x). 11:04, 11 May 2011 (UTC)