Talk:Chaos theory

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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.

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)

Chaos incompatible with astrology

Chaos Theory is incompatible with astrology, since everything that creeps, crawls or flies is subject to ceaseless planetary interrelationships. Astrology is properly the study of interaction of planets with each other, with the inhabitants who crawl on the third rock from the sun being along for the ride.

Weather as an example of Chaos Theory is the most absurd of all, since it has been demonstrated for centuries that astrology, in fact, controls the weather. As the Farmer's Almanac has shown for some two centuries, and before you carp, if a butterfly in the Amazon can create tornadoes in Texas, then why are said tornadoes predictable months or years in advance using astrometeorology? The calculations in the Farmer's Almanac are fully two years in advance and are neither state of the art, nor tuned to specific locales. The Farmers in fact claim to use sunspots, which is false, as McCormack proved more than half a century ago. Aside from the fact that sunspots are under the influence of Jupiter and match its 12 year cycle.

Sources: George J. McCormack (1947), C.C. Zain (1949), Kris Brandt Riske (1997), A.J. Pearce (1911), Sepharial (Walter Gorn-Old, c. 1900), Dr. A. Goad (1690), Claudius Ptolemy (c.150 AD), the Brhat Samhita (Varahamihira c.550 AD). A good number of these are actually available in Google Books. I in fact have all of them except Goad in my library. Not only did all of these forecast weather, except for Varahamihira their techniques largely agree, having built upon each other. Just like any "scientific" discipline. Varahamihira did it not with the 12 signs of the zodiac, but the 27 lunar Nakshatras, which are poorly known outside of India. Included in Brhat Samhita are the methods to forecast for diseases and earthquakes. B.V. Raman (d.1998) and other Indians developed their forecasts from the Brhat Samhita. Raman was well known for his forecasts of earthquakes. McCormack used Goad and Pearce to forecast not only weather, but epidemics and earthquakes. McCormack published his work, at his own expense, for fifty years but was spurned by scientists. They would rather shut their eyes and ears and have "Chaos" instead. Which, as one can read both in the main article, as well as in Talk, is a chaotic mess by comparison to astrology.

The current US heat wave, for example, was forecast, using McCormack's methods, from a retrograde Mars on the MC in New York at the moment the Sun entered tropical Pisces, February 19, 2012, 1:18 am. This is not hard to do, the chart is not hard to interpret, nor was the resulting heat unexpected. Yes, the rest of the world was colder. If you're looking for global warming, you will have to look elsewhere.

It is so rare that facts break upon the helpless ignorance that passes for science. God help you all. Dave of Maryland (talk) 01:29, 25 March 2012 (UTC)

Are you joking? Astrology is a crock of bull. 86.25.212.149 (talk) 16:46, 7 May 2012 (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)

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)

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)

Feedback - a BIG part of chaos theory

I added a link to Feedback which is a BIG part of chaos theory. 71.196.11.183 (talk) 01:45, 20 February 2015 (UTC) pocksuppet