# Talk:Chaos theory/Archive 5

This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |

Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 | Archive 6 |

## Confusing

*Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth.*
I think that this article does not explain well how were all these systems determined to be chaotic.Lakinekaki 00:14, 14 February 2006 (UTC)

- The atmosphere is chaotic because its dynamical behavior is governed by the Navier-Stokes equations, shown to have chaotic regimes (references in Ruelle's book); the dynamics of the solar system is governed by an
*n*-body system of equations, known to have homoclinic tangles and to have a positive Lyapunov exponent under careful numerical simulations (references in Gutzwiller's book); for the Earth's core dynamics, an example is given by the dynamo equation; and for population growth the predator-prey equations are chaotic (most books discuss it). For the economy there is no generally accepted set of equations, but different time-series show predictability and chaotic features. References for the other items can easily be found. – XaosBits 13:50, 14 February 2006 (UTC)

- Thanks a lot for your answers. However, I fail to see how are economic time series tested for chaotic features when definitions in the article apply to systems whose parameters are known, while we do not know parameters for economic systems. This question also applies to the Earth's core dynamics. Lakinekaki

- Economics uses many different types of equations; some are classical differential equations, many are stochastic differential equations. Stochasic diff-eq are ordinary diff-eq with additional noise terms. They ave chaotic behaviours as well. These have been discussed in the Federal Reserve Economic Review (the subscription used to be free, once upon a time). Earth's core dynamics are beleived to be describably by means of fluid dynamics. Certainly, Earth's magnetization reversal time series exhibits chaotic behaviour. 22:11, 14 February 2006 (UTC)

- But stochastic differential equations in economics seems characterises the behavior of the continuous time stochastic process Xt. I thought that stochastic process was different from chaotic one.Lakinekaki

- Economic time-series helped motivate the development of phase space reconstruction techniques. These time-series were assumed to come from low-dimensional chaotic systems. The effectiveness (or not) of the approach is today hidden behind proprietary methods employed by quants in Wall Street (and beyond). There is another aspect in which chaotic dynamical systems are used in economics: iterated games have chaotic dynamics. (Look at Saari's work or at cond-mat/9806359.)

- The dynamics of the Earth's interior is dominated by convective processes, which geophysicists model in chaotic regimes. – XaosBits 12:19, 15 February 2006 (UTC)

- I interviewed for the job of a quant once. I had to bone up on Ito's integral before the interview. One paper, which at first seemed terribly boring, was about drill-bit breakage at a printed-circuit manufacturing plant. On the last page was a graph showing money saved per month by applying the solution to the stoch diff eq for drill breakage. It was labelled in units of $10K/month and I slapped my forehead, "holy shit, this kind of money will pay for a few PhD's in no time." linas 23:13, 15 February 2006 (UTC)

- BTW, if you want to make money working on bios theory, get a job as a quant. These companies are willing to try any kind of math at all, any idea, as long as it can "predict" the stock market. linas 23:17, 15 February 2006 (UTC)

- Thanks for answers and suggestion! Actually, I have a friend who is already trying to use ideas from bios to 'predict' the stock market. I don't know how succesful is he in his explorations. Near future will tell! There certainly are few features in bios that are absent in chaos and that economists could find useful. Since you mentioned money, I will not tell here which are those features! Interested parties can contact me for more details. :O) Lakinekaki

- P.S. my scepticism about the claim that economic series are chaotic seems to be justified! From your answers, it seems more to be assumption than the fact, and more stochastic than chaotic. But that's OK. I'm sure chaos analysis is also useful in analyses of these series. Lakinekaki

- Well really, the entire chaos theory is founded upon the idea, and presumably factual notion that different variables can and will yield different results, even if they are seemingly insignificant to the human eye. This concept applies to every scale... microscopic, atomic, sub-atomic, etc., and so, in the purely physical sense, it can be applied to any discipline. As far as the economy goes, I suppose it depends on your point of view. Economics is, in a sense, a purely abstract concept, whereas chaos theory is meant to be applied to -physical and tangible- instances, so if you are thinking of the economy in an abstract way, then no, chaos theory does not apply; however, in reality chaos theory does apply to the actual economy as it does everything else. As I'm sure you can imagine, even the weather effects the economy.

## Verbiage on Attractors

Why is so much space spent in this article on the concept of an attractor? That's a general dynamical systems concept that doesn't have an inherent connection to chaos. Could it be refactored into a separate article, which could then be referenced when talking about strange attractors?

- There are separate articles onm attractor and strange attractors. Why are they so detailed here, and not referenced?Lakinekaki 04:36, 7 April 2006 (UTC)

## Sensitive dependence

Recent update added:

*Consider the dynamical system defined on the real line by mapping x to 2x. This system has sensitive dependence on initial conditions everywhere ...*

Hmmm ... I don't think this is true. The map *x*->2*x* has two fixed points - a repelling fixed point at 0 and an attractive fixed point at infinity. All initial points apart from 0 head towards infinity. With an appropriate metric (e.g. by mapping projective real line onto a circle) we can even say that any trajectory that does not start at 0 *converges* to infinity. That doesn't look like sensitive dependence to me. Maybe another example would be a better illustration of the point being made ? Gandalf61 17:08, 3 March 2006 (UTC)

- I assumed we were using the usual metric on the real line, and that it did not include the point at infinity. Doesn't this go without saying? Maybe not... if that really seems unclear, I can state it explicitly. --Experiment123 19:07, 3 March 2006 (UTC)

- You're both right. The x->2x is a terrible example; although one may argue it has a positve lyapunov exponent, the system is not a bounded system, duhh, so the positive exponent does not imply topological mixing or even ergodicity, so it sure won't be chaotic. linas 23:12, 3 March 2006 (UTC)

- The point of the x->2x example is exactly what you're saying: it's clearly not chaotic, so sensitive dependence on initial conditions does not imply chaos. That's why it's a good example, not a terrible one! --Experiment123 02:21, 4 March 2006 (UTC)

- I have added the definition of
*sensitive dependence on initial conditions*to the page butterfly effect. It's from Devaney's book. — XaosBits 03:13, 4 March 2006 (UTC)

- I have added the definition of

I'm just a newbie and not a very highly schooled mathematician, but I have two suggestions regarding this quote:

*The systems behave identically only if their initial configurations are exactly the same.*

First, I think it should read "system behaves." It refers to the trajectory after two different initial points, yet within the same chaotic system.

Second, I think that the words "behave(s) identically" is slightly misleading. The word "behavior" seems to invoke a concept of what the system does *after* it has been seeded with an initial condition. I think we should take care to avoid the implication that what follows from two different initial conditions cannot be identical.

Thoughts? --afuturehead 18:13, 26 June 2006 (UTC)

What concerns me about the example x -> 2x is that it is being used to justify the questionable claim that sensitive dependence "in itself is not particularly interesting". This is probably not a fair statement, and seems contradicted by this very discussion. One interesting thing about sensitive dependence is that it depends on the metric being used, which is implicit but unremarked in the definition in butterfly effect. Since the mapping x -> 2x fails to exhibit sensitive dependence when viewed in the compactified line, and indeed is not a chaotic map, this would seem to suggest that definition of "sensitive dependence" should be applied only to bounded metrics. Orbits converging to infinity are not in the spirit of the concept, since these are convergent in the compactification.

Part of the difficulty here is that the article is trying to achieve a non-technical discussion of a topic really requiring precise technicalities to get right. A proper discussion should probably take place within the setting of a compact invariant set -- but these words are not really lay terms. Maybe I'll have a go at it and you can see what you think. --Widenet 02:04, 28 June 2006 (UTC)

## Dissipative

I just googled to find some more info on "chaos theory" in conjunction with dissipative structures. I found an old cached version of this article there, but that info has been deleted from this section, and the old cached version did not say much. Perhaps one of you experts on the area could direct me to a suitable website or book? I have some familiarity with both, but I am looking for some more examples here.DanielDemaret 11:06, 5 March 2006 (UTC)

## Chaos theory and complex systems

I just take a brief look to this topic, and I found something to review about categorization. I only add an external link to Santa Fe Institute, but I thing that the links between "chaos theory" and "complex systems theory" are really deeper and that their topics and sub-topics deserve much more cross-linking (perhaps in a common category) just to make browsing easier (I can't do it by myself, since I am far from competent about and I'm working about "very complex system" of horsemanship...)--Alex brollo 15:39, 9 March 2006 (UTC)

- I beleive that maybe 20 or 30 years ago, "chaos theory" and "complex systems" were almost synonyms. I don't beleive that's true anymore; chaos has a much narrower definition: these days, it tends to focus on
*simple*systems, not complex systems. This is in part needed to make headway: one wants to study the simplest possible system that is still chaotic. By contrast, in the field of "complex systems", one is instead looking for ways of discerning the simple parts of a complex system. These simple parts may, or may not be chaotic. So they'r two differnt fields of study. linas 02:01, 10 March 2006 (UTC)

- This is part of why mathematicians increasingly prefer the term
*nonlinear dynamics*rather than*chaos theory*to describe what they study: it's less likely to be confused with something else, in addition to sounding less metaphysical. --Delirium 08:44, 15 March 2006 (UTC)

- This is part of why mathematicians increasingly prefer the term

## Poll: Should 'See also' section include the link Bios theory or not?

Hello! As a result of prior and ongoing discussions, this poll is to identify a course of action regarding the link on Bios theory in the 'See also' section of Chaos theory article.

At least 2 options are below. Wikipedians can only choose one option and should indicate their preference by signing in the appropriate section with four tildes (~) followed by an optional, one-sentence reasoning. Please only assert positive votes, do not indicate negative ones.

Voting will continue to 1 April 2006 23:59 UTC, but may be extended beyond that if any option does not garner a clear plurality of support.

Thanks for your co-operation! Lakinekaki 18:59, 14 March 2006 (UTC)

### Option 1: 'See also' section should include Bios theory link.

- David Kernow 19:53, 14 March 2006 (UTC). I'd say there needs to be some sort of link, even if to indicate why "bios theory"
*shouldn't*be freely associated with chaos theory. From perusing Bios theory, it seems to me that "bios theory" is a subset of chaos theory, namely all those (models of) instances occuring in biologically-related areas. (Re the article's current {{confusing}} tag, I'll offer to rephrase parts of it if that approach is acceptable.) - — Arthur Rubin | (talk) 18:45, 15 March 2006 (UTC). Bios theory seems to be a fork of a subset of
**chaos theory**. If it's legitimate (which I cannot determine without reading articles in journals I don't have access to), and is to be retained as a separate article, there should be a link.- That's what categories are for. Septentrionalis 20:45, 15 March 2006 (UTC)

- Salix alba (talk) 10:09, 21 March 2006 (UTC). A "See also" link seems to be the optimal solution. Bios theory is distinct enough to warrant its own page as it deals with a specific class of chaotic behaviour. More extensive coverage here would give it undue prominence.

### Option 2: 'See also' section should not include Bios theory link.

- Experiment123 19:09, 14 March 2006 (UTC) "See also" links should relate to topics of sufficient importance to be covered in standard textbooks on the subject.
- WP:DNFT Septentrionalis 20:23, 14 March 2006 (UTC)
- XaosBits 22:32, 15 March 2006 (UTC) I have explained in some detail the issues with Bios theory: the problem in the definition of
*novelty*, one of the main distinguishing feature between chaos and bios; the lack of literature in journals with impact; the lack of mention of Bios theory in Mathematical Reviews in the seven years since the first article; and the problems with the Bios theory entry. These topics are discussed in the Talk:Chaos theory and the Talk:Bios theory pages - Especially in major "subject overview" articles like this one, IMO "see also" should link to related articles as defined by what major textbooks and review/survey papers on the subject consider related. I'm not aware of "Bios theory" having such a status. --Delirium 04:04, 16 March 2006 (UTC)
- I believe the Chaos Theory is a very complex and evasive theory..a theory also known as the butterfly effect, plays a big role in our society..there is much chaos in the world today, and much of this is none to be random..and in the chaos theory, chaos is considered to be random..so i feel there is a connection between these two.. Josh
- Gandalf61 08:25, 18 March 2006 (UTC) Was undecided on this, but seeing the merge proposal appear again has made my mind up.
*Please*don't merge, don't even link, keep Bios theory entirely separate so that Lakinekaki can keep his own private playground and the rest of us don't have to waste our time on what is, at best, an incoherent minority POV, and at worst verges on monomania and trolling.

### Option 3: anyone who likes Bios theory should go and improve it instead of having polls

- Wikipedia is not a democracy. We shouldn't even be voting on this issue. The whole thing smells bad, we've hashed this into little itty-bitty bits on the various talk pages, and absolutely zero effort has been made to fix the dozens of problems that the bios theory page has. Until that page is fixed, we shouldn't even be going here. linas 00:23, 16 March 2006 (UTC)
- Agree William M. Connolley 16:17, 16 March 2006 (UTC)

- Hi guys, I just invested quite a time in improving it. Please tell me what you think. Lakinekaki
- I'm sorry, but the bios article makes no sense. After staring hard at it I can't even tell whether the definitions are supposed to apply to time series or to dynamical systems. For example, the first criterion for biotic motion is sensitivity to initial conditions--ok, so the author is talking about dynamical systems. But the later we see, "the distribution of biotic series is assymetrical [sic]". The more I thought about this, the less sense it made. There may perhaps be something to bios theory, but if so, this article does not do it justice. --Experiment123 22:08, 18 March 2006 (UTC)
- The series generated by these systems have asymetric distribution. What is so nonsensical about that?!? Lakinekaki
**That**requires a sensible definition, also. — Arthur Rubin | (talk) 16:23, 19 March 2006 (UTC)

- The series generated by these systems have asymetric distribution. What is so nonsensical about that?!? Lakinekaki

- I'm sorry, but the bios article makes no sense. After staring hard at it I can't even tell whether the definitions are supposed to apply to time series or to dynamical systems. For example, the first criterion for biotic motion is sensitivity to initial conditions--ok, so the author is talking about dynamical systems. But the later we see, "the distribution of biotic series is assymetrical [sic]". The more I thought about this, the less sense it made. There may perhaps be something to bios theory, but if so, this article does not do it justice. --Experiment123 22:08, 18 March 2006 (UTC)

- Please don't expect me to find answers to all your questions. I wrote Bios theory article as best as I could. I am sure if I spend more time reading the sources that I could make that article even better. However, what is important for me is that it is clear enough and understandable (and I think it is). Reader can get an idea about the subject, and if interested, will go and look for the sources him/herself. Probably some definitions that I wrote are not perfect, if you know how to improve them, please do so. Maybe even bios theory definitions are not perfect, and will evolve further. So what? Even the fundamental theorem of algebra went thru so many revisions by the best mathematicians. That's what science is all about. Wikipedia is about presenting current state of science and its history.

- This quote is from the book
*Chaos and time-series analysis*, Julien Clinton Sprott, Oxford University Press. 2003. (page 104, first paragraph): - Although there is no universally accepted definition of chaos, most experts would concur that chaos is the
*aperiodic, long-term behaviour of a bounded, deterministic system that exhibits sensitive dependence on initial conditions*.

- This quote is from the book

- Although introduction to this article gives few definitions, it is not clear how well are all of them accepted. Lakinekaki 21:04, 20 March 2006 (UTC)

- So I think we see the problem: its not just that you can't spell
*asymmetric*, its that you've got no idea what an asymmetric distribution is. Thats fine, in general, you don't have to know these things. But you *do* need to know them if you're writing articles about them. William M. Connolley 21:53, 20 March 2006 (UTC)

- So I think we see the problem: its not just that you can't spell

- You are twisting what I said. Lakinekaki

- My point is that chaos is not so well defined, and yet, you expect bios to be. In the
*Mathematical bios*paper by Kauffman and Sabelli, it sais:*In contrast to symmetric random processes, periodic processes and chaos, bios generates asymmetric statistical distribution.*They did not define asimmetry there, and I am not going to do it here either. That would be Original Research. Few of you are accusing me of being a troll, and yet, you are behaving like that, not me. Stop bothering me! It's not that you don't understand things, you don't want to understand them. You are trying to dissaprove something, by doing OR, that does not fit your view of the world. I don't care! Things have been published like that, and I'm not gonna fix them for you. Lakinekaki

- My point is that chaos is not so well defined, and yet, you expect bios to be. In the

I think you'll find that Chaos is well defined, but you neither know nor understand the definition. Don't you get bored typing all those colons and long for a fresh start sometimes? Asymmetry: well done, you spelt it right 1 out of 2, thats progress. But you still don't know what it means. Defining what it means in this context would *not* be original research - unless you do what you appear to propose, ie, make the definition up out of your own head. What you need to do is understand what your sources mean by it. And you should do that *before* you write about it. I become more and more convinc3ed that you haven't got a clue what any of this is about, but you just like the words for some reason. Wiki is vast: why not write articles about things you *do* understand? William M. Connolley 19:53, 21 March 2006 (UTC)

- My answer is here, for those who have patience and time to read. Lakinekaki 23:19, 21 March 2006 (UTC)

- I would have no problem linking to Bios theory, if that article were even remotely coherent. --- GWO

## Serious problems concerning Bios theory and User:Lakinekaki

Look out! In Talk:Bios theory, I have just listed troubling evidence suggesting serious problems with the claims made in Bios theory and an apparent conflict of interest on the part of User:Lakinekaki. See also WP:NPOV [[WP:VAIN WP:VERIFY and WP:RS. ---CH 06:36, 12 May 2006 (UTC)

## Merging

I just wanted to voice my opposition to merging bios theory with this article. Bios theory is such a fringe topic, if anything at all, that it does not deserve that much attention in an article on such a broad field as chaos theory. -- Jitse Niesen (talk) 11:08, 18 March 2006 (UTC)

I agree with Jitse Niesen. One link to it should be enough.Lakinekaki

## Wow - what a discussion

Guys, although Chaos and Bios theories may be treated as related they are slightly different in scope, how they are used and their targets. They themselves and articles about them will likely evolve further, possibly away from one another in actual subject area. As such these two articles should not be merged.

Should they be listed in one another's "see also" section? Well, tell me, in general, what needs to be listed in any article's "See also" section and then we'll be able to give an answer. I don't have a clear definition myself, but can say that "See also" entries are as frequently useful as they are not and that is quite OK. It is supposed to be that way because of different interests. "See also" sections frequently include sub-classifications, uses or similarly related articles not listed anywhere else in the article we started from. Chaos theory is used for many things and it does not hurt to mention it. It gives the reader more ways to comprehend why it is there and what purposes it serves. In this regard, I think "Bios" theory *should* be included in the See also section of this article.

"See also" section should also be short. Having too many entries defies its main purpose - giving the reader a finite set of paths to follow up. In that regard, if we have articles A, B and C such that article B appears in A's "see also" list and C appears in B's "see also" list, then we might not need to also list C in A if the list is already long. Example? See, for example, article Bird. One can probably find better examples, but this is the one I just thought of.

Since "See also" of this article is not that long yet, I still don't see any problem in including "Bios" in its "See also" list. When and if it starts growing, this may need to be reconsidered and, perhaps, more infrastructure articles added for this alone - e.g. "List of theories based on Chaos Theory" vs. "List of direct uses Chaos theory", etc. Until then just cool down.

--Aleksandar Šušnjar 18:25, 21 March 2006 (UTC)

- Did you actually read the page on bios theory? Maybe there is such a thing, but the current WP page on it is utter bunkum. Until someone can describe bios theory so that it doesn't sound like pseudoscience, it certainly should not be referenced here. linas 15:49, 22 March 2006 (UTC)

## What shell we do now?

Shell we have the link on Tuesdays, Thursdays and Saturdays, and not have it on Mondays, Wednesdays, Fridays and Sundays? :O) Lakinekaki

- Let's see. You called a poll; it came out 6-3 against you, and your efforts to suggest a merge for your faorite theory are consistently overridden. You now want equal time. No, we shall not violate WP:NPOV#Undue weight.
*Verbum sap. sufficit.*Septentrionalis 05:51, 5 April 2006 (UTC)- It's more like 6-5 (if you count Aleksandar's and my vote), or 8-5 if
*option 3 (improve)*votes are considered*against*. From WP:NPOV#Undue weight:*...Articles that compare views need not give minority views as much or as detailed a description as more popular views...*Noone is comparing views here, or trying to put description of bios in chaos article. We are talking about a link in a*See also*section.Lakinekaki 16:01, 5 April 2006 (UTC)

- It's more like 6-5 (if you count Aleksandar's and my vote), or 8-5 if

- No matter which way you cut it, the majority result of the poll was a decision
*not*to include a link to the bios theory article. Why did you propose a poll if you are not prepared to accept its result ? Are you simply trying to stir up controversy ? Gandalf61 09:05, 6 April 2006 (UTC)

- No matter which way you cut it, the majority result of the poll was a decision

- I am accepting result of the poll. I was just hoping more people would vote and give some option significant majority. This way it's more like
*no concensus*.Lakinekaki 14:48, 6 April 2006 (UTC)

- I am accepting result of the poll. I was just hoping more people would vote and give some option significant majority. This way it's more like

## 3 body and attractors

Did I misplaced the sentence about 3 body problem? Are they behaving as chaotic attractor, or just chaotic? I know that Solar system is *attractor*. Lakinekaki 06:04, 7 April 2006 (UTC)

- The word
*chaotic*often substitutes for another technical term. Different dynamical systems are chaotic in different ways: as Axiom A systems, as mixing systems, as uniform hyperbolic, etc. So when one reads in non-technical discussions that the three body problem*is*a chaotic dynamical system, what is meant is that the phase space of the three body problem is a mixed phase space with regions of periodic behavior and regions that have embedded horseshoes. That means that there is a neighborhood of initial conditions of the three body problem that behave just like a uniform hyperbolic system. So rather than say all this, it is common to say that the system is chaotic. It is similar to usage of*smooth*when referring to differentiable functions. — XaosBits 22:35, 7 April 2006 (UTC)

## Commentary on definition

The text that was moved:

Gleick (1989, p. 306) wrote that *No one [of the chaos scientists he interviewed] could quite agree on [a definition of] the word [chaos] itself,*[1]. That's why he gave descriptions from a number of practitioners in the field some of which are: *the complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems* (refering to "chaotic" behaviour); *a kind of order without periodicity* (refering to chaos), etc.

Textbooks devoted to chaos do not really define the term. Some descriptions include[2]:

*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.*

*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."*

*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.*

*Chaotic behavior can be predicted only if the initial conditions are known to an infinite degree of accuracy, which is impossible.*(Jerry Gollub and Thomas Solomon, Academic American Encyclopedia.)

Rather than to give a precise definition of chaos it is much easier to list properties that a system described as "chaotic" has.

(Moved by XaosBits 22:49, 7 April 2006 (UTC))

- The text that XaosBits has removed is very close to a straight lift from mathworld, and so could be copyvio. If we are going to include a discussion of the definition of chaos, at least let's find our own quotes. Gandalf61 11:05, 8 April 2006 (UTC)

I was asked by Lakinekaki to explain why I moved the text.

The issue of the definition of chaos has been worked out in the literature. As I mentioned above, the apparent difficulty is that the term *chaos* is used, when referring to dynamical systems, the same way the term *smooth* is used when referring to the differentiability of functions. It serves as a shorthand for other technical terms that mean almost the same thing. One author may use the term chaotic for a uniform hyperbolic system, whereas another author may use the term chaotic for a Hamiltonian system that has developed Cantori.

Gleick's book, while a bestseller and helpful in providing an overview of the excitement in the field in the late eighties, is not a mathematical textbook. When trying to ascertain the definition of chaos from textbook, one needs to take into considerations the compromises authors are faced when balancing simplicity of explanation with precision.

Current authors seem to have settled on Devaney's definition, which is the one in the Wikipedia article. In the mid-nineties that definition was simplified in terms of periodic orbits. Devaney's definition focuses on the points of the phase space. If instead the focus is on the measures supported by the dynamics, then two possible definitions for a chaotic dynamical system are common: (1) it has a mixing measure; or the more restrictive (2) it has an SRB measure.

One difficulty does arise when studying systems numerically. It is common to take the concepts that are well defined for an Axiom A system, say the concept of shadowing, and use the concept in a system that appears to be Axiom A. Usually the numerical experiments will show that the concept could be extended to this new situation in some probabilistic sense. The difficult task of making clear the *probabilistic sense* is left for future work. It is a hard task to define the class of systems used in these numerical experiments. — XaosBits 13:08, 8 April 2006 (UTC)

## Newton

Now here I have to disagree. The roots of theories don't originate in solutions to problems, but in the formulation of problems themselves. Henry was solving the problem that dated back to Newton and that others could not solve. Question was asked by Newton, answer was given by Henry. Lakinekaki 15:09, 8 April 2006 (UTC)

- I have not read
*Principia Mathematica*, only little bits, but it does not seem to me that Newton would be discussing predictability. Maybe Lakinekaki could quote. — XaosBits 02:43, 9 April 2006 (UTC)

- Other readers should be aware that many of the historical statements inserted by user Lakinekaki are incorrect. He is prone to reverse wars, which makes it difficult to fix the main text. XaosBits 04:49, 9 April 2006 (UTC)

- This is an accusation. Please be specific which historical statements that I wrote are incorect. Lakinekaki 06:27, 9 April 2006 (UTC)

- I can find no evidence that Newton studied the three-body problem. Wikipedia and scienceworld both agree that the first published studies were by Euler in 1760 and Lagrange in 1772, by which time Newton was dead. I will look through my translation of the
*Principia*to see I can find anything, but I think Newton's character makes it unlikely he would openly discuss a problem that he could not solve. Lakinekaki - do you have a reference ? If we have no evidence, I think it is going too far to say that Newton tried to solve the three-body problem - the most we can say is that he*may*have studied the three-body problem, since it is a natural next step after the two-body problem. Gandalf61 09:43, 9 April 2006 (UTC)

- Here are refs:

*Astronomy & Geophysics*, Volume 41 Page 6.21 - December 2000

*Success and failure in Newton's lunar theory*, Prof. Sir Alan Cook FRAS

- 'Newton's theory of the Moon's orbit is often seen as a failure, but that is misleading. As the first statement of the three-body problem and the first attempt at a solution, it was for the most part remarkably successful. The outstanding defect was the theory of the rotation of the apse line. Halley (1731), towards the end of his life, asserted that Newton's theory was correct to 2 arcmin, and materials of his in the archives of the Royal Observatory, recently analysed, show that estimate to be justified for the period 1725 to 1730 (Cook 1996, 1998). What did Newton achieve in his lunar theory, why did he fail conspicuously in one problem, and why should we still be interested in his achievements?'

- Lakinekaki 15:45, 9 April 2006 (UTC)

- know what guys, do whatever you want. i thought i was improving the article, but you don't seem to like my edits. you can also erase that animated image if you like, and sarkovskii's connection, and Yoshisuke Ueda paragraph, and every other thing i added. i dont' care, you've been quite hostile to me, and i'm not gonna bother any more. Lakinekaki 15:53, 9 April 2006 (UTC)

- Another ref for newton having though about the three body problem is http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Orbits.html from the farly respecatable st-andrews histoy of maths site
*In the Principia Newton also deduced Kepler's third law. He looked briefly (in Propositions 65 and 66) at the problem of three bodies. However Newton later said that an exact solution for three bodies "exceeds, if I am not mistaken, the force of any human mind. "*

- --Salix alba (talk) 15:58, 9 April 2006 (UTC)

- Another ref for newton having though about the three body problem is http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Orbits.html from the farly respecatable st-andrews histoy of maths site

- Thank you - that was exactly what I was looking for. Going back to the
*Principia*, these parts actually make interesting reading. In Proposition 64 Newton solves the three-body problem for the case where mutual attraction is proportional to separation - an academic exercise, since this is not a physically realistic model. Then in Propositions 65 and 66 he talks about the three-body problem for an inverse square law force. In contrast to the very precise geometric proofs in the rest of Book 1, these Propositions are qualitative discussions of special cases - what we would today call "hand waving". Gandalf61 09:31, 10 April 2006 (UTC)

- Thank you - that was exactly what I was looking for. Going back to the

I've reverted to XB. It seems clear enough that whether or not Newton studied the 3-body problem, he didn't study chaos. Identifiying a problem as difficult is quite a different matter. William M. Connolley 16:38, 9 April 2006 (UTC)

## Shall we have a Cultural Refrences Section?

There is a book that has a refrence to Chaos theory. I think it was Jurrasic Park (god I hope I spelt that right). 68.228.33.74 08:00, 30 May 2006 (UTC)

I have created a page about the Chaos Theory of Literature which makes reference to *JP*, in the context of Harriett Hawkins' book *Strange Attractors*. -- Scartol 17:03, 29 June 2006 (UTC)

## Irrelevant links

I removed the following external links, which I considered to be irrelevant:

If anyone disagrees, let me know. If these links are to be used at all, I would recommend putting them in some kind of "list of Nonlinear Dynamics research groups" or something. I don't think they would be immediately useful to someone looking up information about chaos theory. --BryanD 20:04, 6 June 2006 (UTC)

## Chaos for the future

If you could calculate the chaos of one person, could you determine ones future?

- I guess in theory you could, but then again no one knows if chaos theory even exists. That's why it's called chaos
*theory*, not the Law of Chaos or something. It would also be impossible to predict the "chaos" of a person. Chaos is usually figured around things that have set paths to choose. For instance, the water mill example. The water mill, even if it's chaotically rotating from left to right when the water flow is increased to a high amount, the mill only has two directions to turn no matter what. It can only turn left and right, so the chaos of which direction is contained within those two directions. Humans, however, having freedom of choice, always have many options to choose from. For instance, let's say you were in a room and there were two doors leading separate ways. You could go through one of the doors, you could go into a door then come back out, you could use the other door, you could open a door and then shut it without going through it, you could do nothing at all, you could jog in place, etc. As you can see, the possibilities for us are endless, and thus the chaos is not contained by any specific perameters. Thus, even if chaos theory were true, it would be impossible to actually read one's future by determining their "chaos". 24.15.53.225 02:38, 3 June 2007 (UTC)

## Chaos theory and Theory of biological evolution

I have tried to find articles about how Chaos Theory can be applied on Evolution. I wish to use an argument that Evolution by Natural Selection from a Common decent is possible because certain systems in physics bring Chaos from Order as the Double Pendulum. Is this a valid argument?

--Msitua 17:33, 6 August 2006 (UTC)

## Three body GIF

I find that the GIF animation illustrating the three body problem adds very little to the text, so I have removed it. The three body problem played an important role in the development of dynamical systems and it could provide a great illustration. A direct simulation may be interesting, if run for a longer time. This may be impractical with a GIF, but there may be other solutions that do not overload the page. – XaosBits 02:56, 7 August 2006 (UTC)

## The Chaos movie about edward lorenz

Is not about edward lorenz its an action movie...

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