Talk:Chaos theory

Weather and climate

Weather is chaotic. Climate isn't, in general, at least not obviously. So it is a poor example to include here, and unnecessary, so shouldn't be. It looks to me like some of the insistence on including climate is POV-driven (see http://wattsupwiththat.com/2012/01/10/the-wonderful-world-of-wikipedia/) William M. Connolley (talk) 11:31, 12 January 2012 (UTC)

You claim that "Weather is chaotic. Climate isn't, in general, at least not obviously" - without a cite, those statements are OR. The claim that "Climate isn't" is contradicted by the cite that was previously in the article. Cadae (talk) 13:37, 13 January 2012 (UTC)

The OR doesn't matter, since I'm not stating climate-is-not-chaotic in the article. Indeed I wouldn't make such a non-nuanced statement.

Did you actually read the cited articled? Its Sneyers Raymond (1997). "Climate Chaotic Instability: Statistical Determination and Theoretical Background". Environmetrics 8 (5): 517–532. Don;t be mislead by the title, read the abstract [1] William M. Connolley (talk) 13:58, 13 January 2012 (UTC)

My point is that weather is the prototypical example of chaos theory, making it a very good example to include in that list. As said list is going to be non-exhaustive, it seems a bit silly to include climate as well. Can we have an actual counterargument? My apologies if I missed it in the back-and-forth edit summaries the article has seen lately.

Hooray for blogs. Ignoring that but reading the abstract just for fun, the point is that the chaotic input from weather does not necessarily go classical when it becomes climate? Sounds fine as far as it goes, but I am really unclear on why people want to cite a fourteen year old article of dubious relevance. FiveColourMap (talk) 14:43, 13 January 2012 (UTC)

That is my point. My argument for why climate isn't chaotic (at the moment, at least) is [2]. But I'm not suggesting we include that William M. Connolley (talk) 15:01, 13 January 2012 (UTC)

Works for me, thanks. I think there may be a scale argument ("modern" climate since the last ice age vs. predicting longer scale variation), but that is precisely the sort of nuance that I think should be avoided at this article. We have a whole swath of articles to present that material. Since we seem to be basically in agreement here, I have edited the article accordingly. I used Lorenz's foundational paper, along with a more recent book to show a modern perspective. FiveColourMap (talk) 15:51, 13 January 2012 (UTC)

FiveColourMap - thanks for the cites, but they don't appear to be relevant - one is about determinism (which is not necessarily chaos) and the other doesn't relate to climate or weather. Regarding WMC's comment above "Did you actually read the cited articled? ... Don;t be mislead by the title, read the abstract" - the abstract states "Relating the observed chaotic character of the climatological series to the non-linearity of the equations ruling the weather and thus climate evolution". The article points out that the long term (i.e. climatological) data is chaotic. Cadae (talk) 01:12, 14 January 2012 (UTC)

No, it doesn't. And anyway, as FCM says above, we really don't need this kind of nuance on this page. One obscure primary ref does not suffice. The series is climatological, yes. Because it is a 150-y series. The series exhibits chaotic behaviour, yes. But that is not the same thing as climate exhibiting chaotic behaviour William M. Connolley (talk) 16:36, 14 January 2012 (UTC)

People here keep claiming that while weather is chaotic, climate is not. But it's only an ipse dixit. If you have a mathematical proof for that, please provide it in the references. As far as I know, they defined the climate as averaged weather with some complexities added (for details, look up the definitions). It is not true that one can make a chaotic system not chaotic by extending the system to include some more complexities, I'm pretty sure you can figure that out. What remains is that somehow due of averaging, the result is not chaotic. Although this can be true in special circumstances (as in statistical physics, for example, but check out the assumptions, those are not true for the discussed issue), it is false in general. So, if you don't want to have only a religious statement on your hand, please provide some proof (a mathematical proof would be nice). One can find references that the climate is chaotic (even IPCC acknowledged some 'components' http://www.ipcc.ch/ipccreports/tar/wg1/504.htm), for example here: http://onlinelibrary.wiley.com/doi/10.1002/joc.632/pdf "For example, the climate

system is currently modelled by systems of coupled, non-linear differential equations. Chaotic behavior is the prime characteristic of all such systems. This results in unpredictable fluctuations at many time-scales and a tendency for the system to jump between highly disparate states. It is not yet known if chaos is the primary characteristic of the climate system but the Earth’s climate has been documented as undergoing very rapid transitions on time-scales of decades to centuries (Peng, 1995 and Figure 2). There is no reason to believe that this characteristic will disappear in the future."

And please add back my edit about the measurement errors. They can be way bigger than rounding errors in computers.

I understand that the one that usually edits out the 'climate' (and also removed the mentioning of measurements errors) was a climate modeler. Looks like he might be biased. — Preceding unsigned comment added by 79.119.58.201 (talk) 07:56, 12 September 2013 (UTC)

Weather and climate II

Isn't the point of chaos theory that there is no chaos? It is a euphemism that points out our inability to see complex patterns. And by complex I'm talking predicting the place and vector of any atom in a glass of warm milk. Chaos theory says it can't be done and I agree. But not because it is impossible but because we are incapable. The wingflap of a butterfly *does* set off a tornado in Texas but we will never(?) be able to point a finger at the animal and say: "She did it." Or more spesific: "She will do it and...that was the flap." --94.212.169.79 (talk) 10:07, 26 September 2012 (UTC)

Indeed. "Chaos Theory" is one of the biggest misnomers in the history of science, since what it studies really isn't chaos at all, but simply another kind of order (nonlinear order). "Imaginary numbers" are also a misnomer too, since they aren't really imaginary (as those of us who have studied quantum mechanics know). LonelyBoy2012 (talk) 21:04, 25 December 2012 (UTC)

Fractals and Bifurcations

Wouldnt be worthy mentioning this? Most chaotic attractors have fractal properties and there's a huge number of cases in which chaos can arrise from parameter perturbation such as Feigenbaum cascades and Shilinikov chaos? — Preceding unsigned comment added by Lbertolotti (talk • contribs) 19:44, 19 February 2013 (UTC) --Lbertolotti (talk) 19:46, 19 February 2013 (UTC)

HorseShoe Map

One of you smart people out there has got to be informed about the horseshoe map im talking about. I saw a picture of it once and had a brief explanation that left completely lost. I've look around the internet and can't find any reference of it. Essentially the concept is related to topology, it involved a process of folding a rectangle in a repeatative fashion that left it in the shape of a horseshoe. It was intended to show sensitive dependance on initial conditions. Two points that begin close to each other could end up far apart — Preceding unsigned comment added by 173.166.29.105 (talk • contribs)

(moved from the article page William M. Connolley (talk) 20:56, 1 March 2013 (UTC))

Errm, did you mean Horseshoe map? Its, ermm, linked from the article William M. Connolley (talk) 20:58, 1 March 2013 (UTC)

Catastrophe Theory

In the history section, some mention of catastrophe theory is needed, since a lot of it can be considered the precursor to modern chaos theory. — Preceding unsigned comment added by 24.17.185.145 (talk) 22:40, 27 March 2013 (UTC)

Minimum requirements for chaos

The article states: Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it has to be either nonlinear or infinite-dimensional.

However I believe periodic forcing in a linear system can create chaos. I saw this in a lecture by Dr. Robert L. Devaney of Boston College. Putting a spring in a box and shaking it can cause chaotic behavior. — Preceding unsigned comment added by 99.111.104.36 (talk) 15:54, 24 August 2014 (UTC)

Nothing is chaos

is this really passing for science/mathematics? I'd vote to have this article removed. There is no such thing as chaos, nothing happens for no reason or out of order. If something happens there is a cause for it to happen. If you bounce a ball and the ball behaves a certain way, but you bounce it the same way as far as you can tell and it bounces different, then there's simply a calculation you are missing (the spin of the ball, temperature, static in the air etc) there is always a cause for an effect, to say otherwise is simply a chaotic statement. 50.47.105.167 (talk) 17:54, 27 May 2013 (UTC)

Chaos does not simply mean something happens for "no reason." It means the slightest of changes can cause great change. Models of weather, for example, give significantly different predictions when even a rounding error is made. That means to predict the weather, we would have to know were every molecule involved in weather is. That is what is meant by saying weather is chaotic. Other things, like say, baking cake, are not chaotic. Putting in slightly more less than the recipe calls for causes only a slightly different cake. TheKing44 (talk) 18:03, 27 May 2013 (UTC)

I think I understand better now, so this theory does not rule out the cause-and-effect law, I misunderstood the theory as to mean literal "impossible to determine" while it may be impossible with current science, I'm sure in the future better tools would be able to make better predictions. 50.47.123.176 (talk) 18:47, 23 July 2013 (UTC)

No, you did not misunderstand. Improved measurement accuracy increases the time predictions can be considered useful, but "chaos" would eventually occur. — Arthur Rubin (talk) 16:35, 13 June 2014 (UTC)

I think this is actually quite an interesting question, as the anonymous poster obviously has some wrong ideas of what science and maths do, and those ideas are quite likely widespread, but I find it hard to pin them down. Their views also appear to clash with quantum indeterminism. They do not seem to appreciate that chaos means that in the end no approximation is good enough: if I understand right this means that no margin of error on the initial state can rule out reaching any other state being reachable in the long term to within the same margin. That is an attempt to reformulate the conditions for chaos (mixing, dense periodic orbits) in less technical terms while still conveying their force, but I think it can be improved.

PJTraill (talk) 08:51, 26 August 2014 (UTC)

Distinguishing random from chaotic data

Is wrong. It can converge exponetialy to 0 and also computation looses precision.

-Comment added to article by 79.117.14.226 (talk), 18:16, 9 December 2013

I'm not sure what the above comment means but the section "Distinguishing random from chaotic data" does look in need of improvement. Yaris678 (talk) 19:02, 9 December 2013 (UTC)

This is a better phrase all round: "The chaos equation cannot be solved, but it can still be useful." — Preceding unsigned comment added by 86.163.193.74 (talk) 10:15, 8 March 2015 (UTC)

Change of sources by an IP

I'm not keen on this edit, which replaces one source with another. The previous source wasn't the highest quality, but I think it was sufficient for the purposes we used it for. The new source is available on Google books, and I can't find the quote mentioned in it.

The IP has made other, valuable edits to the article, but I don't know where this has come from. Am I missing something?

Yaris678 (talk) 18:00, 9 July 2014 (UTC)

I've had no response so I have reverted the change of source. The source that the IP cited was a self-published book that doesn't appear to contain the quote. Yaris678 (talk) 13:07, 9 September 2014 (UTC)

Rigorous definition: sensititivity to initial conditions, discrete/continuous

The definition of Sensitivity to initial conditions is not as rigorous as the other two — can that be improved? The lack of rigour resides in “significantly”, in “each point … is arbitrarily closely approximated by other points with significantly different … trajectories. Thus, an arbitrarily small change … of the … trajectory may lead to significantly different … behavior”.

I suspect that the reason could be either that this condition is generally only used in informal definitions (since it is redundant, at least some of the time) or that different people use different definitions of “significant”, but it would be nice if someone could clarify this. It sounds a bit as though the Lyapunov_exponent might be useful for a stricter definition.

The section Topological mixing gives exponential growth as an example of sensitivity without chaos, but even (increasing) linear growth has the property that “any pair of nearby points will eventually become widely separated”! Perhaps they can be distinguished by a suitable definition of “significantly different trajectories”?

I also note that some example systems proceed in discrete steps, while others (e.g. the jointed pendulum) are functions of real-valued time: the definition should perhaps clarify if both are permitted. I suppose that follows from the definition of a dynamical system (which article also does not specify it), but it might still be helpful to mention it here. PJTraill (talk) 22:50, 28 July 2014 (UTC)

I agree on the point about "significantly". I can imagine a more rigorous definition, based on any achievable distance from any point... but as Wikimedians we summarise other people's work, rather than developing or own, so it would be better to find a source that gives a better definition of sensitivity to initial conditions.

As you (and the article!) point out, this part of the definition isn't actually necessary. Perhaps one approach we could take is to move the words on sensitivity to initial conditions to a different/new section. Leaving the definition to be based on the more rigorous stuff.

Yaris678 (talk) 15:09, 1 August 2014 (UTC)

Jerk systems

There is a section entitled such in the artcle Jerk (physics), which, imho, does not really fit to the physical content of that page. It just refers to the third derivative motivating the name from kinematics. Recently, I did some work on that physics page and would like to shift this content here, where, if I do not mistake this matter, it would fit better and were appropriate also. Certainly, it would require some adaptation to a more mathy lingo, and there are already simpler circuits published, with only one diode as non-linearity, but the discussion on in some sense minimal systems appears to me sufficiently interesting for this page. May I, please, ask for comments. Purgy (talk) 10:06, 16 August 2014 (UTC)

I did as announced above, and hope, not to have deteriorated something.Purgy (talk) 10:24, 20 September 2014 (UTC)

Chaos And Computation

The finding that universal computation would be almost surely chaotic is debated upon. I am the author of the paper, and after the paper went to press, they notified us saying other people have found flaw in the proof. In the light of the flaws therein ( unless we manage to hold our position ) the citation or argument should be removed.

• • Furthermore, it has also been argued that universal computation is 'almost surely' chaotic.[73]** — Preceding unsigned comment added by Nmondal (talk • contribs) 01:49, 3 September 2014 (UTC)

Done. — Arthur Rubin (talk) 09:17, 9 September 2014 (UTC)

Apparently the argument has been won : http://www.sciencedirect.com/science/article/pii/S0304397514005222 The paper is published, and therefore, anyone else trying to add the link and the statement back should be fine. Please let me know if anything else is required. — Preceding unsigned comment added by Nmondal (talk • contribs) 18:02, 25 September 2014 (UTC)

Order & Chaos definition

Hi, after writing a theory of interaction proposal, I set to consideration two new definitions to order & chaos. I think that current definitions are completely wrong. Please see the introductory videos about interaction and dimensionality on ydor.org. Over that basis:

• The Systems Theory is an objective approach of nature; but the Theory of Interaction is a subjective approach of systems, the modelization of how systems approach other systems in nature; science will not be able to understand natural systems until observing them from the interactional point of view.

• A dimension is an approach of contents processing (see the video). The redefinition of dimension permits applying the same rules of mathematical systems interactions to real systems, natural systems. A system of equations has a complete different set of dimensions than a natural system; notably, natural systems have compound complex dimensions; then:
  • example 1: approach: compare a distance against a ruler; contents: two points; processing: measuring; that is a linear distance, a 1-dimensional space;
  • example 2: approach: extracting nutrients by digestion; contents: milk; processing: drinking; that is the output content of a cow system, milk, from the subjective point of view of a human; the complex & compound dimension called milk.

• Interaction is the mechanism of exchange of subjective of dimensional contents between systems that causes a profit value.
  • example 1: exchange: two atoms exchange stability by fundamental interactions; profit value: increase of the scale of existence; example of subjective dimensional content: the exchanged force.
  • example 2: exchange: cow gives milk to farmer, farmer gives pasture; profit value: positive for the farmer, drinks & sells milk, positive for the cow, continues living)

• Order is the dimensional disposition that exists during interaction;
  • example 1: H2O is 3 atoms holding repetitive interactions. The dimensional disposition could be the 104.45deg or the 95.84pm.
  • example 2: To speak (interaction is speak over the air) with someone, you need to be @ 1m distance, no obstacles, etc. After speaking, order is lost.

• Chaos is the lack of interaction on a dimensional space. Order coexists with chaos on different dimensions.
  • example 1: if the H2O molecule breaks, positional order is finished (order on the positional dimension). If the molecule was moving, the particles could

keep the same energy after splitting (despite there is chaos in positions, speeds continue to be the same; in other words, there is order on the energy dimension, but chaos on the positional dimension).

Rodolfoap (talk) 22:45, 20 March 2015 (UTC)

While Wikipedia does occasionally allow some authors to cite themselves, it does not allow self published books to be cited. I see that your work also cites Wikipedia, which is problematic in that it could result in circular sourcing. Wikipedia is not the place for you to promote your ideas. Go find an academic publisher and some peer-reviewed journal, get published in there, then your ideas might be presented. Ian.thomson (talk) 23:15, 20 March 2015 (UTC)

Spontaneous Order? --> Simplexity

The small section of spontaneous order if badly flawed. "Spontaneous" itself is part of the problem. If it is deterministic (as chaos is claimed to be) it cannot be spontaneous. Just because a set of cyclic phenomena occasionally appear to display coordinated does not mean there is any order. Set three lamps blinking at different rates. Occasionally two will flash at the same time. Rarely, all three will flash at the same time. There is no order here, there is only the initial disorder progressing as it was programmed to, and the asynchrony of the initial conditions produce an illusion of synchronization. There is no order here, any more than every tornado is Texas can be blamed on a Brazilian butterfly. Nor is there any spontenaity. The eventual coincidence of flashes was predictable when the flash periods were chosen.

The inclusion of neurons as examples is particularly egregious. The examples given in the referenced text are of artificial neural networks. While these produce results that appear similar to the action of living neural nets, such artifices operate on principles entirely different from the outcome models. Neurons do not spontaneously synchronize. For example 85% of the human brain performs inhibitory action. One such inhibitory action is pulses of transmitters such as GABA injected into collections of neurons, such as cortical pyramidal cells, each with its own spontaneous firing rate. The inhibitory pulse delay neural firing that was about to happen. Those that were closest to firing are delayed most. Their firing is pushed back until they begin to coincide with those just slightly behind the first in time. This repeats until the entire collection (ie. Hebbian cellular assembly) is firing together. The inhibitory pulses continue until that particular assembly is no longer needed for the task at hand (or some are called into action as members of other assemblies). The most obvious supposed synchrony was also the first EEG ever seen -- alpha waves. These occur not as a resting state but when enough of the local neural population (52% or more according to Nunez) is operating on a single task. It occurs when the eyes are closed, not because the cortex is taking a break, but because it's seeing a single thing -- the darkness behind the eyelids. The same result is obtained when the eyes are kept open but covered with halves of ping pong ball. All white or all dark field of vision doesn't matter. All the same does. This is not spontaneous, there is a very specific cause, and a very specific mechanism that provides that cause for a very specific kind of neural processing task. — Preceding unsigned comment added by Drmcclainphd (talk • contribs) 11:47, 30 March 2015 (UTC)

Wikipedia only cites mainstream academic sources without interpretation or elaboration, and only modifies articles accordingly. If you cite some sources and propose specific changes (e.g. "change X to Y because it's in line with Z source"), you'll find that the article is more likely to change. A wall of text without citations will have about has much effect as saying "the article is wrong." Ian.thomson (talk) 18:36, 28 April 2015 (UTC)

History?

This article does an admirable job of explaining what chaos theory is, but doesn't make it at all clear where it came from, other than a brief mention of Lorentz. When was chaos theory first propounded or proposed as "chaos theory"? By whom? Was it Lorentz? Did he publish his proposal somewhere? How was it received? Etc.

I'd suggest taking a look at the Quantum mechanics article. The last paragraph of the introductory section gives a concise history of the development of the theory; something like that is needed here. — Preceding unsigned comment added by 74.95.43.249 (talk) 18:26, 28 April 2015 (UTC)

Also: Minor detail, but do we want to say, at the bottom of the history section, that the journalist James Gleick "upheld" the thesis that Chaos Theory constituted a "paradigm shift" in the Kuhnian sense? He's a masterful journalist, but not a divine oracle. As Kuhn points out, it takes a few generations—or in case of relativity superseding Newtonian physics, a few decades—in order to make sense of the messy hurly burly of day-to-day science. Instead point out that Gleick "agreed" with the thesis. (Which, not for nothing, isn't really a theory in the definitive sense, just the colloquial one, but now I'm splitting hairs.'' --Jeffreyphowe (talk) 21:48, 28 July 2015 (UTC)

This article's title is an example of total nonsense.

There is no such thing as chaos theory.

Chaos-related concepts are part of the theory of dynamical systems. It is in no way a field or subfield in mathematics. Although popular writers — who get all their information from other popular writers — use this term, that does not make it part of mathematics. It makes as little sense as claiming that the study of the number π is a subfield of mathematics.

Chaos, although it lacks one single widely accepted definition, is nevertheless a concept studied in dynamical systems, and merits its own page just as many other mathematical concepts do. That does not mean there is such a thing as "chaos theory".

The title ought to be changed to either Mathematical chaos or Chaos (mathematics).

It would be much better if articles on mathematics were written by knowledgeable people.Daqu (talk) 15:33, 11 May 2015 (UTC)

You may have noticed that most of the 57 Wikipedia's use a similar title. So I guess you assume that also all those users are not knowledgeable. Bob.v.R (talk) 05:53, 14 May 2015 (UTC)

You might have noticed also that over 80,000 scholarly publications use the term. I suppose you would claim those academics are also not knowledgeable. By contrast, hardly any academics use the term mathematical chaos --Epipelagic (talk) 06:07, 14 May 2015 (UTC)

Like it or not, Epipelagic's links demonstrates conclusively that the term "chaos theory" is very much used and "mathematical chaos" (~1,120 results) is not. It does, however, strike me that the examples that come up first for "chaos theory" are mainly from somewhat softer disciplines: medicine, economics and life sciences in general. Even restricting it to "chaos theory"+mathematics (~35,800) or "dynamical systems" (~19,800) seems to yield a similar bias. The results for "mathematical chaos" do include Douglas Hofstadter, who is well known, and they show a similar bias. So one question is what do its practitioners call it? My impression from the references in Chaos theory itself is that plain "chaos" is most popular, with "chaos theory" second. A second point is, that if we ignore Daqu's strange and unhelpful polemic, the suggestion of Chaos (mathematics) does seem consistent with other mathematical topics. PJTraill (talk) 13:51, 14 May 2015 (UTC)

Just in case anyone failed to notice: A lot of people use a lot of words and phrases that have no actual referent. The fact that that word "yuppie" was used countless times in the early '80s was not slowed down by sociological studies showing that No, there was no new demographic category that was suddenly beginning to grow at that time.

What I said was that there is no discipline called "chaos theory", and I stand by that. The relevant discipline is called "dynamical systems". The word "chaos" — which has several inequivalent definitions that are currently used — is a characterization that applies to some dynamical systems and not others.

There is no "theory" called "chaos theory", regardless of how many times or places that phrase is used. There. Is. No. Such. Thing.

People can cite all the ignorant references they want, but that does not make "chaos theory" into a real thing. (The word "theory" implies that it is a discipline. It is not.)

Maybe instead of calling what I wrote "strange" and "unhelpful" and a "polemic", detractors who have nothing to say but pejorative words — without addressing even one thing I wrote — would please sit down and stop soiling the pages of Wikipedia.

It doesn't matter how many times the phrase "chaos theory" is used. Uneducated persons who do not know much about what they are writing use the term only because others have used the term.

It has been written in many books about mathematics that the first uncountable infinity is that of the continuum. This appeared in George Gamow's book "One, Two, Three,...,Infinity" and was repeated in many, many other writings about infinity. It is false — or more accurately has been proven to be independent of the axioms of set theory.

Also, virtually every other mistake that has appeared in Wikipedia for any length of time can be found in countless other writings, since the Internet is like an echo chamber. That is why we have to be unusually careful about what we put in articles here, that many readers will unfortunately take as Truth.Daqu (talk) 07:47, 25 May 2015 (UTC)

This article should be moved

Chaos may be the lack of interaction or the lack of a pattern, but in any case, it is not possible to study something that does not exist (how to study the things that have ended existing?) or to study something that is the opposite of something positive like order (how to study all non-mouse things?). Thermodynamics is a good effort to study dissipation (and dissipation itself is a type of order), but that's it. Once chaos rise (you do not interact anymore with your dead friends), how can we study it? There is no study of chaos.

Another important thing: A lot of physical dynamics as fractals, attractors, complex motion patterns are understood as chaos. Probably we call that chaos because we don't understand it and we don't understand order. But that is the subject of the complex systems theories. Complex systems generate complex patterns, but any logic mathematical proposition is never an example of chaos. Those are just complex patterns. Please stop calling that chaos. That is not chaos at all. If you disconnect gravity and connections from the double rod pendulum, parts will be expelled from the model, that is chaos.

Plase classify this article as inappropriate. Rodolfoap (talk) 07:49, 3 January 2016 (UTC)

I see nothing "inappropriate" in this article. Its title may not fit to the most elaborate standards in scientific precision, but I consider it as "not bad" with respect to generating satisfactorily hits for a large group of users.

To no extent I object to any improvement of this article, which contains already yet a considerable amount of valuable information. Purgy (talk) 13:55, 3 January 2016 (UTC)

Dr. Gomes's comment on this article

Dr. Gomes has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

This article provides a good and balanced description of chaos theory. It explains that chaos is a deterministic phenomenon, that sensitive dependence on initial conditions is a central feature of chaotic systems, it makes an important reference to strange attractors and jerk systems, it mentions the main authors responsible for the development of the theory, it distinguishes between continuous-time and discrete-time chaos, and it refers to various applications in distinct fields of science.

I would like to make a single remark basically about the references concerning economics. The article just mentions three articles by the same author: C. Kyrtsou. There are many other relevant contributions relating the application of chaos theory to economics. I mention a few: • Baumol, W. J. and J. Benhabib (1989). “Chaos: Significance, Mechanism, and Economic Applications.” Journal of Economic Perspectives, vol. 3, pp. 77-107. • Boldrin, M.; K. Nishimura; T. Shigoka and M. Yano (2001). “Chaotic Equilibrium Dynamics in Endogenous Growth Models.” Journal of Economic Theory, vol. 96, pp. 97-132. • Brock, W. A. and C. H. Hommes (1997). “A Rational Route to Randomness.” Econometrica, vol. 65, pp.1059-1095. • Bullard, J. B. and A. Butler (1993). “Nonlinearity and Chaos in Economic Models: Implications for Policy Decisions.” Economic Journal, vol. 103, pp. 849-867. • Day, R. H. (1982). “Irregular Growth Cycles.” American Economic Review, vol. 72, pp.406-414.

• Deneckere, R. and S. Pelikan (1986). “Competitive Chaos.” Journal of Economic Theory, vol. 40, pp. 13-25.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Gomes has published scholarly research which seems to be relevant to this Wikipedia article:

• Reference : Orlando Gomes, 2007. "Imperfect Demand Expectations and Endogenous Business Cycles," Money Macro and Finance (MMF) Research Group Conference 2006 127, Money Macro and Finance Research Group.

ExpertIdeasBot (talk) 12:41, 7 June 2016 (UTC)

Draft outline

There's a draft for an outline on chaos theory at Wikipedia:WikiProject Outlines/Drafts/Outline of chaos theory if anyone is interested. -- Ricky81682 (talk) 06:42, 24 June 2016 (UTC)

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External links modified

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Interconnectedness

A quick google search show how Interconnectedness is the primary aspect of chaos theory, hence, I added to the article thusly... 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there is interconnectedness, underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. 73.46.49.164 (talk) 17:14, 21 November 2017 (UTC)

I do not know which results of a quick google search you refer to, but the interconnectedness you linked to is in no way a primary aspect of the topic this article on a more mathematical than interdisciplinary theory is about. I am re-reverting your edit to await consensus in this dispute here on the talk page. Purgy (talk) 17:35, 21 November 2017 (UTC)

You're wrong. Are you a moderator? I AM an expert in chaos theory and of course, it's mathematical and interdisciplinary. Did you bother to google: interconnectedness chaos theory? https://www.scribd.com/document/214467264/Chaos "Concepts in Chaos. Chaos refers to an underlying interconnectedness that exists in apparently random events. Chaos science focuses on hidden patterns, nuance, the..." I'll see if I can find my copy of the James Gleick book Chaos - I AM sure it's also in there. Are you an atheist? Atheists usually oppose interconnectedness because it equates to GOD. 73.46.49.164 (talk) 20:50, 21 November 2017 (UTC)

If you ARE an expert of some chaos theory of your kind that might be reasonably connected to interconnectedness as is explicated in that article, and this theory is notable and second sourced, then you certainly should devise a WP-article on this topic. It is, however, obvious to me that this here article on chaos theory lacks not only any connection to your linkage of interconnectedness, but also lacks almost any connection to your claim of randomness —it's about being deterministically chaotic— and it certainly has no whatever connection to any concept of GOD.

It does not matter whether or not I am a moderator, an atheist, or just a simple editor, it is up to you to organize a consensus about me being wrong here, and your claim —to which I strongly oppose!— that this here chaos theory article is reasonably connected to the linked content via interconnectedness is right. BTW, I do not consider scribd as a reliable source, and I do not oppose to interconnectedness per se. Purgy (talk) 10:41, 22 November 2017 (UTC)

Wrong. Interconnectedness and an appearance of randomness is part of chaos theory. Every believer in GOD believes that everything is connected even though there may be an appearance of randomness. You appear to be an atheist and let your beliefs negatively affect your science. I have yet to find my copy of James Gleick's Chaos so I can use that as a reference. You don't like that the source I provided confirms "underlying interconnectedness in apparently random events', so you claim it's "unreliable". You'll probably let your bias negatively affect any source I provide. 73.46.49.164 (talk) 19:20, 13 December 2017 (UTC)

There is no randomness here, anyone's (dis-)belief in GOD is of no concern here, and it's not about me (dis-)liking something. It's to me obvious that you are mistaking the content of this here article. Please, read and understand the whole article, and that it's about something else, as you seem to assume. There is definitely absolute disconnectedness between your perspective on chaos and this article. Please, write a separate article about your chaos with connectedness. Purgy (talk) 21:34, 13 December 2017 (UTC)

@Purgy Purgatorio: I've strucked the edits above as they were by a sock of Brad Watson, Miami. For other edits by his socks seeWikipedia:Sockpuppet investigations/Brad Watson, Miami. Doug Weller talk 10:52, 15 April 2018 (UTC)

I've tweaked "interconnectedness" to "interconnection" - it's better English. 2601:582:C480:BCD0:9883:1BAF:3F46:6C2B (talk) 13:01, 19 July 2022 (UTC)

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Critics section must be added

The Chaos "theory" is not theory at all. It is a product of the weaknesses of the mathematical tools we use to model real world processes. That we cannot yet find suitable mathematical means for a more accurate description of these natural phenomena does not give legitimacy to the crippled tools of calculus to create such shits as "Chaos theory". I look at this article more like yours "Alchemy" one, than of scientific section from Wikipedia. Probably not just me. A stable critical section must be added necessarily.

P.s. I've seen other people to criticize you above, too.

"Small differences in initial conditions, such as those due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general."

It is strange how exactly "rounding errors" lead Edward Lorenz to this shit, but yet many people still take this fabrication for reality. The quality of Wikipedia articles has begun to fall under all criticism. You even consider tabloid information to be accurate, just because many people have distributed it. I think that with the gentleman in question above we are wasting our time. https://www.imdb.com/title/tt0387808/

Emil Enchev 84.238.148.54 (talk) 11:54, 20 December 2019 (UTC)

Thank you for your suggestion. If you provide links to reliable sources of such criticism (satisfying WP:RELIABLE and WP:VERIFY), somebody could integrate them into such a section, in the spirit of WP:NEUTRAL. If not, it is not so likely to happen. I also notice you refer to "you", but you would be welcome to make that "we" if you were prepared to work in accordance with the Five pillars. PJTraill (talk) 19:26, 22 December 2019 (UTC)

Chaos Theory - An Interdisciplinary Theory

I improved the opening paragraph... Chaos theory is an interdisciplinary theory and branch of mathematics focusing on the study of chaos: dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions.<ref]"The Definitive Glossary of Higher Mathematical Jargon — Chaos". Math Vault. 2019-08-01. Retrieved 2019-11-24.</ref><ref>"chaos theory | Definition & Facts". Encyclopedia Britannica. Retrieved 2019-11-24.</ref] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. 99.169.79.198 (talk) 11:01, 5 September 2021 (UTC)

Chaos as a Topological Supersymmetry Breaking

I included a small subsection mentioning that dynamical chaos and its stochastic generalization can be viewed as a spontaneous breakdown of topological supersymmetry. Even though this viewpoint on dynamical chaos is relatively new, it has already been published in a dozen scientific peer-reviewed journals and article collections including Physical Review, Annalen der Physik, Chaos etc. Therefore, it is not an "original research" for Wikipedia regulations and it clearly passes "reliability source" criterion.

It is understood that this wikipage is mostly for general public. However, Wikipedia is also used for scientific work to look things up quickly. I hope this new subsection will serve as an occasionally useful cross reference for such visitors.

Generally speaking, chaos and supersymmetry are certainly among the most fundamental physical concepts and there are a few good reasons why to mention this newly established relation between them on this wikipage. For example, the topological supersymmetry breaking seems to be the only mathematically rigorous definition of dynamical chaos, at least to the best of my knowledge. The traditional trajectory-based approaches did establish most of the important and definitive properties of this phenomenon but they failed to unravel its very essence from the mathematical point of view. This is actually the reason why professional mathematicians in dynamical systems theory try to avoid using the term chaos. Thus, the topological supersymmetry breaking is the only existing solid link between mathematics and the term chaos, which is basically somewhat of a misnomer as follows from the next paragraph.

For another reason, this new understanding reveals that dynamical chaos is a low-symmetry, or "ordered" phase as a theoretical physicist would recognize it. This corrects a very common misunderstanding that dynamical chaos is some sort of randomness -- a point of view which is in clear contradiction with the fact that dynamical chaos exhibits infinitely long memory of initial conditions, perturbations etc.

Please help improving this subsection by direct edits or discussions here.

Vasilii Tiorkin (talk) 06:12, 5 April 2022 (UTC)

A broad article like chaos theory should contain the topics that are in every standard text on the subject, not new perspectives that are still clamoring for attention. Wikipedia is not the place to promote new ideas that we happen to think are deserving. XOR'easter (talk) 13:35, 5 April 2022 (UTC)

While the related web page did survive an AfD in 2017 (although I'm not entirely sure how), I'm not convinced that this belongs in the main Chaos theory article, and I'm SURE it doesn't belong there in its current form. The article as a whole is a pretty solid chunk of writing on a technical subject for a non-technical audience. In other words, it reads like an encyclopedia article should.

The deleted paragraph, on the other hand, would be incomprehensible to the average reader. I have a PhD in physics, and I BARELY followed it. If the topic can't be explained in sufficiently clear language to match the level of the rest of this article, then it's probably too niche to be included, and at best belongs in the "see also" list at the bottom.

That's independent of the notability concerns raised by @XOR'easter, which I think are also serious. PianoDan (talk) 16:42, 5 April 2022 (UTC)

Wikipedia regulations particularly state ("Wikipedia:What wikipedia is not"): ... If you have completed primary research on a topic, your results should be published in other venues, such as peer-reviewed journals, other printed forms, open research, or respected online publications. Wikipedia can report your work after it is published and becomes part of accepted knowledge; however, citations of reliable sources are needed to demonstrate that material is verifiable, and not merely the editor's opinion. This certainly applies in this case because the material has been published in multiple reliable sources including Chaos, Phys.Rev. E/D, Modern Physics Letters etc. The editors and reviewers of the above journals have agreed that it is true. I honestly do not understand why you guys want to label this material "original research" in the Wikipedia sense. The way I see it, it is not fair Vasilii Tiorkin (talk) 04:13, 6 April 2022 (UTC)

I investigated this topic. There are a number of primary papers, but I didn't see any published review papers on chaos and TSB specifically. Probably the best way forward is to create an article on the topic, perhaps analogous to supersymmetric theory of stochastic dynamics based on secondary reliable sources. If one can create such an article (it might be too soon for these sources to have been published), the article may be worth a link or a mention in this broad article. --{{u|Mark viking}} {Talk} 16:57, 5 April 2022 (UTC)

Perhaps this paper https://www.worldscientific.com/doi/abs/10.1142/S0217984919502877 entitled "Chaos as a Symmetry Breaking Phenomenon" may partly serve the purpose of a review of the relation between Chaos and TSB. Vasilii Tiorkin (talk) 04:42, 6 April 2022 (UTC)

I don't see anyone invoking WP:OR in this discussion. Rather, the argument is that this particular theory hasn't established that it is of sufficient importance to include in the main chaos theory article yet. PianoDan (talk) 16:51, 6 April 2022 (UTC)

I cannot think of anything more important for "chaos theory" than to finally get an exact definition of chaos. Such definition did not exist before. TSB does exactly that. In fact, TSB does more. It also generalizes chaos to stochastic models, which are more accurate models of real systems because the later always experience influence from noise. It reveals the physical essence of chaos (spontaneous breakdown of a symmetry is a very general and fundamental phenomenon) and shows that it is actually an "ordered" or low-symmetry phase and that the corresponding order parameter is fermionic. It makes direct connection to the topological field theories -- a very important class of mathematical models etc. In other words, it links the "chaos theory" to a bigger cluster of mathematical and physical knowledge which on its own may result in fruitful crossfertilization of different scientific disciplines. Roughly speaking, it makes the mathematical apparatus of high-energy physics applicable to, say, neurodynamics or stockmarket. This will certainly lead to a few interesting findings in the future. As to the present, I have very little doubt that TBS picture of chaos is important and thus notable.Vasilii Tiorkin (talk) 01:33, 7 April 2022 (UTC)

"Important" and "Notable" are VERY different things in the context of Wikipedia. And hard as it is to believe, we simply don't care about the former. I could create nuclear fusion in my garage tomorrow, and that would be hugely important, but until it's covered in the secondary literature, it wouldn't be the least bit notable.

It can be tough to accept, but whether or not a theory is CORRECT just isn't the main issue - the question is whether the theory has widespread enough NOTICE to include in an encyclopedia. And for a top-level article like this one, the bar for the AMOUNT of notice it has received is higher than for an article specifically on the theory itself.

If this theory is as important as you say, it will eventually be covered in secondary sources - textbooks, review articles, and other secondary sources. And when that coverage reaches a reasonable level, it will warrant inclusion in this article. But I don't think it's there now. PianoDan (talk) 04:08, 7 April 2022 (UTC)

I realize I said "importance" in my initial reply when I mean "notability," and I apologize for the confusion. PianoDan (talk) 04:09, 7 April 2022 (UTC)

About WP:OR issue above: Sorry, I should have mentioned earlier that the predecessor of the deleted subsection was tagged by an anonymous editor as "original research" and later deleted because particularly of "dubious sources". Both are false judgements of course. That is why I had to address it.

On the notability, what you are saying does sound right even though I do see a potential contradiction with the Wikipedia rule highlighted in bold above, unless of course Wikipedia has some additional rules for top-level articles.

Either way, I am no expert on the Wikipedia publication rules but I would find it very unnatural if Wikipedia regulations were designed in a way that prohibits a top-level wikipage referencing a lower-level wikipage just because of a notability concern. If this is right, what would be the right way to make such a reference in this particular case ? To me, "see also" does not feel quite enough because the subjects of these wikipages overlap so strongly. On the other hand, I was told :) that it is too early for the TSB picture of chaos to have a separate subsection on this top-level wikipage. But maybe the reference on it could be an short integral part of the narrative, like a sentence with a link in it ? Vasilii Tiorkin (talk) 04:36, 8 April 2022 (UTC)

I don't think there's any concern with linking to more specific pages, just with which material is WP:DUE in the general one.

Let me build an example: let's say I came up with a brand new map projection tomorrow.

If I were to immediately create an article on WP for it, that would be WP:OR.

On the other hand, if I were to publish it in a specialist journal, and maybe get a few citations in other specialist journals - now a case can be made that it is no longer OR, and you have to look at WP:GNG and WP:PSTS - it still doesn't have any coverage in secondary or tertiary sources, so is it has to make a very strong case based on just primary sources. A lot of specialist math and science topics fall into this pile. At this point, my projection might be worthy of a page, but probably not a mention in the main text of the "map projections" article, and DEFINITELY not in the main text of the "Geography" article.

If after a few years, there's an article in National Geographic about the projection, and it starts to be widely adopted, well - NOW we have a very strong case that the projection is sufficiently notable not just for its own page, but for mention in the "Map Projections" article. If it were to become a standard projection used widely, it would probably even rate mention all the way up in the "Geography" article.

And the thing is - absolutely NONE of these criteria have anything to do with how GOOD the projection is. That's completely irrelevant. The only thing that matters is how much it is covered and used. Wikipedia is based on the assumption that coverage in (particularly secondary) external sources is the metric for notability.

So in this case, I think if you could write a single sentence description that was accessible to a lay reader who had read at minimum, the portion of the article preceding it, you'd have a stronger argument that it shouldn't be reverted. But I think that's about as far as this should go in this article right now, and I'm still not sold on that. I certainly think there's no shame in adding the link to the sub-page from the "Other Related Topics" section at the bottom. PianoDan (talk) 17:02, 8 April 2022 (UTC)

I did not manage to find a good place for a sentence yet but I will keep looking. For now, I just added a link in other related topics.

Thank you very much for your helpful explanations. Vasilii Tiorkin (talk) 18:52, 8 April 2022 (UTC)

In accordance with our discussion, I have added a sentence at the end of the "Spontaneous Order" section.Vasilii Tiorkin (talk) 02:10, 27 February 2024 (UTC)

Boolean algebra and trigonometry, partitions and permutations

How they work,how to solve them , require various example of each.. Hasheela William (talk) 05:25, 8 June 2023 (UTC)

Contradiction: which is weaker?

"Topological mixing (or the weaker condition of topological transitivity)"

...

"Topological transitivity is a weaker version of topological mixing."

Which is it? GreatBigCircles (talk) 15:34, 18 October 2023 (UTC)

Chaos theory should be labeled as pseudoscience

This "theory" only shows the immaturity of the mathematical tools used in discrete computer modeling of real continuous processes. Chaos theory is nothing more than pseudoscientific propaganda. — Preceding unsigned comment added by Emil Enchev BG (talk • contribs) 14:00, 28 March 2024 (UTC)

Your opinion is irrelevant. Wikipedia is based on reliable sources, not on what you think. --Hob Gadling (talk) 10:47, 24 August 2024 (UTC)

"Disorganized" listed at Redirects for discussion

The redirect Disorganized has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 8 § Disorganized until a consensus is reached. Duckmather (talk) 06:22, 8 April 2024 (UTC)

Photo/ Video example is misleading

The video that is replicating a supposid chaos theory example is misleading as it is apparent upon viewing the video that the hand is providing different momentum to each segment of the six segment video. This seems more as a misrepresentation. B1blazin (talk) 07:46, 13 April 2024 (UTC)

"Butterfly effect" is not just sensitivity to initial conditions

The article doesn't describe the butterfly effect correctly. Lorenz actually meant something stronger than initial condition sensitivity alone. Palmer et. al. 2014 describe it like this:

Historical evidence is reviewed to show that what Ed Lorenz meant by the iconic phrase ‘the butterfly effect’ is not at all captured by the notion of sensitive dependence on initial conditions in low-order chaos. Rather, as presented in his 1969 Tellus paper, Lorenz intended the phrase to describe the existence of an absolute finite-time predicability barrier in certain multi-scale fluid systems, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times.

For example, the famous Lorenz 63 system does not have the butterfly effect. Its initial condition sensitivity is continuous, so the forecast error can always be reduced by improving the initial state estimate.

Palmer, T. N., Döring, A., & Seregin, G. (2014). The real butterfly effect. Nonlinearity, 27(9), R123. https://doi.org/10.1088/0951-7715/27/9/R123

Abstractchaos (talk) 03:57, 30 June 2024 (UTC)