Talk:Chaos theory/Archive 2

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Removed section

I took these few sentences out of an early part of the article, it was more dense than even I could get through. I think this passage as it currently reads may be too technical for the encyclopedic level we're generally shooting for, I'd like to work on understanding what it's saying and finding a way to re-word it in a way that is more clear and understandable. Any thoughts? I also added some comments into the text itself (look in the edit window to see)Ëvilphoenix ^Burn! 07:45, 20 December 2005 (UTC)

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Sensitivity to initial conditions is related to the Lyapunov exponent.

Transitivity means that application of the transformation on any given Interval $I_{1}$ stretches it until it overlaps with any other given Interval $I_{2}$.

Transitivity, dense periodic points, and sensitivity to initial conditions can all be extended to an arbitrary metric space. J. Banks and colleagues showed in 1992 that in the setting of a general metric space, transitivity and dense periodic points together imply sensitivity to initial conditions.

This elementary but unexpected fact prompted Bau-Sen Du, of the Institute of Mathematics, Academia Sinica, Taiwan to define a stronger version of sensitive dependence - extreme sensitive dependence - which is not a consequence of transitivity and dense periodic points. Extreme sensitive dependence means, roughly, that points close together separate and converge infinitely often, as is often the case in examples of chaotic dynamical systems.

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I'm gonna leave it here for a while, I'd love some input on this! Ëvilphoenix ^Burn! 07:45, 20 December 2005 (UTC)

I agree with most of these changes, which definitely improve the article. I have reversed one change in the intro, where I felt that even though was more appropriate than however, and I have re-introduced the explanation of transitivity, with an attempted expansion and re-wording. Gandalf61 09:55, 20 December 2005 (UTC)

Ok, great. You added the following sentence:

Topologically transitive means that system will transform any given interval I₁ of its phase space until the image of this interval eventually overlaps with any other given interval I₂.

Ok, so there's some things in this sentence I'd like to better understand:

1. what is phase space, and what is an interval in phase space?

   Ahh, there is an article, I'll add the wikilink into the main article and study that to try and understand that better.

   Is there an article that could be linked that discusses the term interval as it is used in this context?

2. what is meant by the image of the interval?
3. what does it mean for intervals to overlap?

Thanks. Ëvilphoenix ^Burn! 19:58, 20 December 2005 (UTC)

I won't argue to much with this edit, since its poorly written, although we need an article on this topic. I strongly disagree that its "too technical", I understood it just fine. I strongly disagree with the general statement "too technical for the encyclopedic level we're generally shooting for"; the removed text is less technical than just about any article in the Category:Mathematical analysis or Category:Topology or Category:Geometry; in other words, its less technical than 70% or 80% of the math articles on WP. linas 14:41, 20 December 2005 (UTC)

The fact that this might be less technical than other math articles does not mean that it's still not too technical itself...the other articles themselves might also be too technical. The idea is that a reasonably average layperson should be able to read the article and have a reasonably decent understanding of what is being explained. The encyclopedia is intended to be an introduction to the topics it discusses. That being said, in my mind theres nothing wrong with discussing advanced concepts, but I think at the very least the first few paragraphs should be comprehensible, and that's mainly what I'm shooting for. Something that will help that is explaining some of the terms used in the article. While they are wikilinked to other articles that can provide a more in depth explanation, it would be good to have a brief explanation of complex terms in the article itself when they're introduced, and that's one thing I'm trying to do to make the article more accessible. I'm glad you were able to understand that text that I removed, may I consult you with any questions I have on clarifying it? Ëvilphoenix ^Burn! 19:45, 20 December 2005 (UTC)

OK, I agree that much of this article should be accessible to the average college-educated layperson. However, most math articles will never be accessible to the average layperson, unless that person has a at least a major in math, and in many cases, a PhD in math, specializing in that particular area. Again, just try randomly browsing the categories I just mentioned, and you'll see what I mean. However, I think we are in agreement that many key articles, such as this, should be at least partly accessible to all. I didn't mean to scare you or bite, its just that the label "too technical" sometimes gets thrown about with little respect; good editing is better than shallow accusations.

To answer yourquestion, yes, you can ask, and I'll try to answer. User:Gandalf61 is good at this stuff too, better than me, in fact. linas 02:08, 21 December 2005 (UTC)

Thanks, and I will ask. Nice edits to phase space, by the way. I'm still waiting on a response from Gandalf61 on my question above, I'd value your input too. You didn't scare me and I didn't feel bitten, I'm too experienced on here to be easily intimidated (or intimidated at all, really), and you're more than free to disagree with my opinions, but I'm certainly not lightly calling the article too technical, I'm willing to work and make it better. I still think all articles on the encyclopedia should be accessible, but I'm too lazy to look up the relevant policy/guideline/whatever it is that's guiding that opinion right now. However, yes, this article at least should be accessible, and this is where I'm focusing my efforts. I actually think with some better citations this could be a good FA candidate. Ëvilphoenix ^Burn! 05:31, 21 December 2005 (UTC)

What's the question again? I think you are looking for interval (mathematics). If f is a function (mathematics), then f(x) is the image (mathematics) image of a point x, similarly for intervals. I'm going to redirect topologically transitive to Mixing (mathematics) becase I think that is the intent of this section. You might find mixing to be "too technical". linas 00:16, 22 December 2005 (UTC)

Perhaps Mixing (physics) is the less technical presentation of almost the same thing. linas

Thanks, I'll study those articles. Ëvilphoenix ^Burn! 03:00, 22 December 2005 (UTC)

topologically transitive

I beleive that a system is "topologically transitive" if it is topological mixing. Is this right, or is there a looser/different defintion for topological transitivity? (I'd like to cange to article to say mixing instead, if its agreed). linas 00:26, 22 December 2005 (UTC)

BTW, note the sentance "There are examples of systems that are strong mixing but not topologically mixing, and examples that are topologically mixing but not weak mixing." in that article. I don't know if chaos theory requires a particuar type of mixing, or whether more loosely "any kind of mixing will do" for a system to be chaotic. (Personally, I think "any kind will do", but maybe someone out there is more strict). Qualify: "any kind of mixing that is stronger than ergodic" linas 00:29, 22 December 2005 (UTC)

Yes, topological mixing matches my understanding of "topologically transitive", and it also has the advantage of being a more intuitive term than "transitive". Gandalf61 09:24, 22 December 2005 (UTC)

Biotic motion

(New discussion moved from top of page)

I removed the biotic motion link. As far as I can tell, biotic motion is a neologism used to describe the type of behavior Sabelli and Kauffman see in the iteration,

A_t+1 = A_t + g sin(A_t).

Sabelli and Kauffman (in Mathematical bios, Kybernetes, volume 31, page 1418) define bios as a pattern resulting from a nonstationary aperiodic series that satisfy certain properties that are not standard in the ergodic theory literature. Maybe someone could define bios precisely. XaosBits 02:37, 13 January 2006 (UTC)

I too removed the biotic motion link because I can't see how "biotic motion" is anything more than a special type of standard chaotic behaviour. Lakinekaki re-instated the link, so I asked him on his talk page how "bios" differs from chaos. I have reproduced his reply below. Gandalf61 09:54, 13 January 2006 (UTC)

Gandalf61 - I can say few things about chaotic and biotic motion: first, chaotic motion is bounded to one or more attractors, and as is written on chaos theory page, it has dense trajectories (by the way, there is an error on chaos page - chaotic motion is not necessarily periodic as is described in chaotic motion section). This is not true for biotic motion. In mathematical sense, you can measure a few factors to distinguish between the two. In chaos, any two consecutive data points can be anywhere within the range of the series. In bios, this is not true. Bios is more "continuous". Also, by doing quantitative recurrence analysis, chaotic and biotic series give qualitatively different results caused by different natures of their motions.

"What, in mathematical terms, characterises the transition from chaos to "bios"?" An "escape" from an attractor. (as I see it) Also, paper you read is kind of old, there are more recent papers about the methods used in distinguishing between chaos and bios. Lakinekaki

From what I read last night, biotic motion seems to be a random walk where the steps of the walk are given by a chaotic dynamical system. XaosBits 12:21, 13 January 2006 (UTC)

"random walk where the steps of the walk are given by a chaotic dynamical system" (chaotical systems are deterministic) - isn't this an oxymoron, random walk generated deterministically? Lakinekaki 22:47, 13 January 2006 (UTC)

Hi, FWIW, the equation

A_t+1 = A_t + g sin(A_t).

Is known as the circle map. It was studied by Yakov Sinai and Kolmogorov in the 1950's and 1960's, it appears in Kolmogorov's fat multi-volume series of that era. I had a good friend review it in his PhD thesis in the 1980's, and I have pictures of it on my personal art gallery, dating back to the early 1990's. I'll try to write a WP article on it, but it will take me maybe a month to track down the refs and review what is generally known to distinguish from my own personal research. linas 15:25, 13 January 2006 (UTC)

I should mention that it is "well known" that the circle map describes the behaviour of certain phase locked loops (PLL's), including both the mode-locking regions (aka "Sinai's tongues"), and the regions where the PLL's go chaotic. Insofar as the human heart vaguely resembles the PLL, it should not be surprising at all that "biotic" heart-beat intervals resemble the behaviour of a PLL in its chaotic regime. In particular, the article on PLL's gives the equations for a PLL, and the circle map was a highly simplified model of the PLL because Kolomogorov wanted to simplify the PLL equations as much as possible in order to study them. And yes, he says so, alluding to both the electronic circuits and mechanical systems (a rotor coupled by a spring to a motor -- again a heart-like mechanical system). linas 15:44, 13 January 2006 (UTC)

Hi all,

People like to quote Wikipedia here when deleting things. Let me remind you of some of Wikipedia's basic policies:

Wikipedia should only publish material that is verifiable and is not original research. One of the keys to writing good encyclopedia articles is to understand that they should refer only to facts, assertions, theories, ideas, claims, opinions, and arguments that have already been published by a reputable publisher. The goal of Wikipedia is to become a complete and reliable encyclopedia, so editors should cite credible sources so that their edits can be verified by readers and other editors.

'Verifiability' in this context does not mean that editors are expected to verify whether, for example, the contents of a New York Times article are true. In fact, editors are strongly discouraged from conducting this kind of research, because original research may not be published in Wikipedia. Articles should contain only material that has been published by reputable or credible sources, regardless of whether individual editors view that material as true or false. As counter-intuitive as it may seem, the threshold for inclusion in Wikipedia is verifiability, not truth. For that reason, it is vital that editors rely on good sources.

Wikipedia:Verifiability is one of Wikipedia's three content-guiding policy pages. The other two are Wikipedia:No original research and Wikipedia:Neutral point of view. Jointly, these policies determine the type and quality of material that is acceptable in the main namespace. The three policies are complementary, non-negotiable, and cannot be superseded by any other guidelines or by editor's consensus. They should therefore not be interpreted in isolation from one other, and editors should try to familiarize themselves with all three.

If you think that certain authors disagree with biotic motion, then you should add that to the line I am adding, but do not delete things ignoring wikipedia content-guiding policy. Your disagreement with something published in the paper cannot eliminate those facts from wikipedia - encyclopedia. Those papers have passed peer reviews, and contributors on this forum should not do the job of peer reviewers.

Linas also mentioned here PhD received in 80's assuming (in my oppinion) that I shall be intimidated by credentials. Louis Kauffman is professor of mathematic at University of Illinois at Chicago at this moment, and if I am about to reason on credentials, he may know more what is happening in recent years in the field then a person who did simmilar research in 80'.

Linas also said that A_t+1 = A_t + g sin(A_t) is known as the circle map. Can you show me the equivalence?

http://www.ceptualinstitute.com/genre/sabelli-kauffman/bios.htm is a reference I put on Bios theory page and it is well explained there what is the difference between the two equations above (for those who want to read).

It is possible that I did not explain well the difference between chaotic and biotic motions. I just started an article, and I am certainly going to add much more details to it, but it's incompletness cannot justify your ignorance. If you think that explanation is not good enough, you may want to help me editing Bios theory article. That would be much more constructive, and I would certainly appreciate it.

Lakinekaki 16:17, 13 January 2006 (UTC)

Please do not strike out on a warpath. Wars leave only casualties, and no winners. Some remarks:

• It would be much more constructive if WP newcomers did not attempt to quote WP policy to long-time WP editors.
• WP is an encyclopedia, and not neccessarily a place to report new research results, or advocate new theories. The neologism "biotic" will not be found in most published books reviewing chaos theory, and that alone should give you pause. There may well be authors that use that term, but it appears to be new and is clearly not widely accepted.
• I am not trying to intimidate you. People who know what they are talking about are not usually intimidated when thier knowledge is challanged.
• You ask Linas also said that A_t+1 = A_t + g sin(A_t) is known as the circle map. Can you show me the equivalence? I don't understand the question; equivalence to what?
• Please don't call anyone ignorant. I don't care how smart you may think you are, calling other people ignorant is rude.

linas 22:51, 13 January 2006 (UTC)

response to linas

WP is an encyclopedia, and not neccessarily a place to report new research results, or advocate new theories.Where did you read this guideline? How can you judge what does encyclopedia 'neccessarily' encompase. This results may be few years new, but they are published! Accept that fact.

The neologism "biotic" will not be found in most published books reviewing chaos theory, and that alone should give you pause. There may well be authors that use that term, but it appears to be new and is clearly not widely accepted.What is your argument here, we shell read on wikipedia only what is widely accepted? Also, this is not about chaos theory, it's about bios theory - no wonder you couldn't find it in the chaos literature. Also, one book and one paper is enough if it is rescpectable Journal. Are you questioning the reputation of the Journals cited?

I was asking about equivalence between circle map and

 At+1 = At + g sin(At) 

Is this equivalence still in question? I don't want to clutter this page if I don't have to. Septentrionalis 03:29, 18 January 2006 (UTC)

Please don't call anyone ignorant. I don't care how smart you may think you are, calling other people ignorant is rude.I apologize for that, I just didn't perceive it as a rude word. It was not my intention to be rude.Lakinekaki 23:01, 13 January 2006 (UTC)

Please stop taking such a belligerent attitude in your posts. There is an article at MathWorld on the circle map, at http://mathworld.wolfram.com/CircleMap.html -- perhaps it may be able to answer your questions. See also Standard Map linas 23:26, 13 January 2006 (UTC)

I have not found a mathematical definition of biotic motion. The papers cited in the bios theory article fail to give a mathematical definition of biotic motion. The other concepts expounded in the Chaos Theory article are all backed by rigorous and mathematical publications. Many Wikipedians have worked hard to explain in non-technical terms the technical concepts of dynamical systems. As far as I can tell (as I have not found the definition) biotic motion is a diffusive process where the steps of the walk are given by a chaotic system. In the example of the process equation, after a time dependent change of variables, the iteration of the process equation asymptotes to a random walk with step size following a 1/|cos| distribution. I suggest the link be removed and maybe incorporated into a See Also section. XaosBits 05:00, 14 January 2006 (UTC)

Dear Lakinekaki, please note that what Sabelli and others call process equation (with g not dependent on time) is known in the dynamical systems literature as the lifted circle map with bare rotation of zero. The lifted means that the domain of the equation is the reals and not the circle. When g depends on time, it is no longer a circle map. XaosBits 05:07, 14 January 2006 (UTC)

Thank you XaosBits for pointing out that I should put some mathematical definitions in the Bios theory page. I found definition for novelty, and will add more definitions till monday evening - have plans for this weekend! And just one note to your comment about biotic motion being a diffusive process - that is not true. It may and may not be. There are both diffusive and non-diffusive biotic processes. Also, biotic motion and bios theory are not just about the behaviour of this specific equation, but about the features that characterize whole class of processes and equations. Some equations may have been analyzed in the past, but the fundamental distinctions have not been found until recently. Lakinekaki 22:31, 14 January 2006 (UTC)

I have again removed "biotic motion" from the list of dynamical system behaviur types. I have no problem with having a link to bios theory in the See also section of the Chaos theory article. What I disagree with is the attempt to promote biotic motion to a new type of behaviour, distinct from chaotic motion. I have seen nothing so far that justifies this view. Biotic motion is a sub-type of chaotic motion which arises from a particular family of maps. Gandalf61 15:13, 14 January 2006 (UTC)

How are these things on the image the same? Could you explain it to me please? How come, by your logic, that chaos is not just sub type of bifurcating/periodic behavior? What do you have in mind when observing this image that makes you claim the equivalence of the motion before and after the transition from chaos to bios. I ask for an explanation. If you do not provide it within few days, I will put biotic motion link again.Lakinekaki 17:58, 14 January 2006 (UTC)

The explanation is quite straightforward. The bifurcation diagram above is for an initial point in the range [0,2π]. If you choose an initial point in the range, say, [2π,4π] you would see the same pattern of bifurcations shifted vertically by 2π. In other words, the full bifurcation diagram shows an infinite series of identical bands with vertical period 2π. When g is less than a threshold value each interval [2nπ,2(n+1)π] is mapped to a subset of itself, and the map has an infinite series of strange attractors. When g exceeds this threshold the strange attractors merge and the Julia set of the map in now the entire real line. The threshold value occurs when the maximum value of x+gsin(x) for x in [0,2π] exceeds 2π. A little calculus and 5 minutes with a spreadsheet shows that this threshold value is approximately 4.6033388487517. There's nothing fundamentally new happening at this threshold value - the domains of chaotic behaviour simply merge. It's still plain old ordinary chaos. (Just noticed that :XaosBits has posted essentially the same explanation further below - but it was interesting working it out for myself).Gandalf61 14:30, 16 January 2006 (UTC)

I believe the above image is a cross-section of the image below. The period-doubling part is slicing through the Arnold tongues (the black V-shaped areas) and the diffusive part is a slice through the chaotic parts (the colored regions). If I understand it correctly, the slice is the along the vertical line exactly in the middle of the picture. I'll have to think about this some more, its intersting.

linas 19:45, 14 January 2006 (UTC)