Talk:Chaos theory/Archive 3

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

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

Archive 6

Redux

Hi all!

I was thinking about this discussion today, and have realized that it was probably my fault why you didn't understand my arguments. I did not look on this from the perspective of a the person who is knowlegable about chaos, but for whom the bios theory is totaly new, and yet ressembles so much to chaos. I will try now to make up for that and give you better explanation of the origin of this theory. Why I acted in a way I did is probably because (I am human too!) few people provoked me by deleting biotic motion link without any real argument, and to that behavior I refered as ignorant - which is, by the way, not a rude word, but describes an attitude of a person in a certain moment (see ignorance). I probably acted the same, by not trying hard enough to understand what you do not understand. However, few arguments and questions (especially by XaosBits about mathematical definitions) made me think more about this (and add some mathematics to the page!), and now I will try to give an explanation as best as I can. Chaos has been studied for decades, so has chaotic diffusion. But never earlier have been these two found together in the same equation in such an obvious way as they have appeared in the process equation. From the image above that I put, you can (or maybe not so clearly :) see that equation goes thru many stages: convergence to steady state, a cascade of bifurcations, chaos, bios and infinitation (not visible on the image), according to the value of the feedback gain g. I can understand that before this observation, chaotic diffusion was considered just that - a chaotic diffusion. But by observing this equation's behavior, it become evident that this diffusion deserves a distinct name. Otherwise, chaos could be called for instance just bifurcational mixing where bifurcations stop being distinguishable and their number reaches a very high value. Also, following same reasoning, bifurcation could be called steady state separation! (try to apply now this to definition of chaos.) Sometimes new concepts are needed, and therefore new words. (remember that what used to be random 150 years ago, become chaos 100 years ago.) Kauffman and Sabelli realized this. So they published the paper. So they can be cited in encyclopedia. Now, please tell me if there is still something that I could explain better, I will certainly be happy to try. Lakinekaki 07:09, 15 January 2006 (UTC)

Let me write here again that what characterizes bios is not diffusion but the generation of novelty, which is absent in non-diffusive chaos but is present in non-diffusive bios! (it is independent from diffusion) Lakinekaki 07:28, 15 January 2006 (UTC)

The problem with biotic motion is not that its may or may not be a new type or motion. There are an infinity of types-of-motion and the difficulty is characterizing one of these well enough. That requires developing a method that distinguishes the motion from other types-of-motion and discovering a few interesting results about the new type-of-motion. For example, there was once just diffusion, where the average position squared grows proportional to the time. Today anomalous diffusion is recognized, where the average square grows with time to a power other than one. There are results related to anomalous diffusion, such as renormalization group calculations of the exponents and predictions of anomalous diffusion based on stability of trajectories. For biotic motion there is no characterization of the type-of-motion, and consequently, no collection of results. In my view, an up-to-date encyclopedia should at most have a modest entry in biotic motion stating it is work in progress.

There have been no review articles in biotic motion. The publications have been limited to proceedings, non-mathematical journals or journals like Kybernetes, with a very low impact ratio (that means, not well cited). Until the topic is a little more mainstream, I feel an entry on biotic motion should be modest so as not to confuse readers that are not familiar with the evolving research. XaosBits 15:25, 15 January 2006 (UTC)

Dear XaosBits, there have been review articles on biotic motion. Although you may not consider Kybernetes, Cybernetics, and other cited journals to have a high impact ratio, we can discuss that issue and analyze in more detail what Wikipedia considers reputable journal. I think that entry on biotic motion is modest - it deals with definition of this motion (and I think that actually it needs more information exactly for the reason not to confuse readers, because at this point, I obviously did not explain well some things there), and I don't see how additional information can confuse readers, as if they do not have critical powers to distinguish between concepts. This topic may not be mainstream, but it is not original research either. I think that if people become aware of this topic, it will actually help it become mainstream, OR in the case that some of your (peer review like) arguments have foundation, it will help this topic go into oblivion (if it happens that someone publishes a paper where he/she shows how this bios can is actually just diffusion thing or what ever might be the argument, etc). I cannot see how doing this is a bad thing. As I said before, there was a time when everything unknown was random, then one day, some of the random things become chaos. Nobody considered it a bad thing! Lakinekaki 16:00, 15 January 2006 (UTC)

Let me try to restate the problem that Xaosbits, and I, and others are having. There is a claim being that biotic motion is important; so important that it should be understood as a new paradigm for the understanding of the motion of dynamical systems. There are several problems with this claim. First, there is no evidence presented that biotic motion is important. If it was important, there would be books on it, review articles on it. If it was a recent, brand-new breakthrough, it would be the topic of seminars and colloquia across the country. There's no evidence that this is happening. I monitor the seminars on chaos at a major university here, (U Texas at Austin) with a large department in these kinds of studies (Center for Non-Linear Dynamics), and there has not been the tiniest hint of knowledge or interest in this topic.

Another problem is that biotic motion is poorly defined. Personally, I am used to very rigorous definitions: for example, see the article mixing (mathematics), which defines five different kinds of chaotic motion. Is biotic motion one of these kinds? See, for example, measure-preserving dynamical system as a rigorous definition of a class of systems that exhibit chaotic motion. Are biotic systems of this type, or are they different? In these discussions, all that I've gotten is the claim that biotic motion is different, but without any clear explanation of how it is different.

The third problem is that the prime example of biotic motion seems to be a slice through the Standard Map, with some of the parameters varied adiabatically over time. Since the standard map is both well-known, heavily studied, and still full of many deep mysteries and surprises, it would be much more comforting if this example of biotic motion was compared to what is already known about the standard map, so that we could tell apart the new results from the parts that are "well known".

Until these three problems are resolved, I believe that biotic motion, although interesting and worthy of mention, should not be elevated to the status of a new paradigm of dynamical motion. linas 17:54, 15 January 2006 (UTC)

First you talk about importance. Could you give me a link to Wikipedia policy that talks about importance of facts in its pages? For my friend Johny Smith Junior none of these is important, bios or chaos for that matter. (If you really want to know my personal oppinion why bios is important, here it is: chaos concept cannot distinguish between living natural process and mechanical ones. Concept of bios makes this distinction. Studies have shown that many complex living processes have features of bios - i.e. generation of novelty, while most mechanical systems lack this feature and are chaotic.) Than you talk about poorly defined biotic motion and you give example of rigorous definitions of chaotic motion. Chaotic motions were analyzed for more than 50 years before chaos name was used and some strict definitions of chaotic motions were defined. If you think that definition of novelty that characterizes bios is a poor definition, why don't you publish a paper on that subject, and then you will be able to add your views from that paper in Wikipedia. What you see as the third problem I don't percieve as a problem. What you think is a slice through the Standard Map I challenge you to prove it (because what you say is not true) by plotting it in the time domain. If you think there are problems to be resolved, you can resolve them in the academic field by publishing papers. None of you gave me an answer to the question about the image I posted here. Do you personally really think that chaos and bios (transition is at the x-axis value of around 4.6) on that image are the same types of motion? Lakinekaki 19:17, 15 January 2006 (UTC)

It is not the case that chaotic motion was analyzed for a long time before being well defined. It started well defined. The work of Poincaré described in very precise terms recurrence and the tangles of the stable and unstable manifolds. In the early 1900s, Duhem used some earlier work of Hadamard on flows on hyperbolic surfaces to show that there could be problems with long term predictions in classical mechanics. The work of May and Feigenbaum, that ignited the chaos work in the mid 1970s, was also mathematically clear. If there is a definition of biotic motion it should be stated in these discussions to short circuit many of the arguments.

The impact of a journal is a number computed by Thompson, the company that provides the Science Citation Index. It is the ratio of citations per published article (in the last two years). Great journals will have an impact ratio of one or more. In mathematics the citation rate tends to be lower, but even then the Kybernetes rate of 0.185 is low. The journals that are most similar to Kybernetes are INT J SYST SCI (impact of 0.239) and FUZZY SET SYST (impact of 0.734). Most similar is computed through a co-citation measure.

None of this means that biotic motion is not a worthy area of study or that it wrong. It is just that in a mature field, such as dynamical systems, the onus is on the discoverer to explain how the work fits in with the rest of knowledge. What is its measure-theoretical characterization? How does the stability of points and orbits relate to the motion? Etc. None of this seems done. XaosBits 20:15, 15 January 2006 (UTC)

Dear XaosBits, what I was trying to say was that by looking at the current definition of chaos at the Wolfram research website [1] or the Chaos theory page, I do not think that the same definition was valid 100 years ago, and yet the chaotic behaviour was analyzed even back then. At this time, what distinguishes bios from chaos is for example mathematical definition of novelty. (You may not like it, but that is your thing.) This does not mean that tools applied in analyzing chaos cannot be applied in analyzing bios. On the contrary.

You have an oppinion that the Kybernetes rate of 0.185 is low. That is fine. Here [2] you can see that some other people consider Am. J. Pharm. Educ. with rate of 0.29 a well recognized Pharmacy Journal. Here [3] you can also see that CYBERNET SYST has a rate of 0.581. But all this thing about rating of the journal is beside the point, and even irrelevant. No wikipedia policy talks about this.

What is its measure-theoretical characterization? How does the stability of points and orbits relate to the motion? You are again trying to do the job of peer reviewers. This is not the place for that. I think that such conservative reactions like yours have led to rejecting Nobel class papers [[4]]. Out of about 500 Nobel prize winners (peace and literature excluded), about 10% had their papers rejected (before they won prizes). More than half of these people were rejected papers for which they won prizes later!)

I find it really interesting how none of you are giving me an answer on chaos and bios in the image above :) Lakinekaki 01:43, 16 January 2006 (UTC)

I cannot comment because I do not know what biotic motion is. I would like to know, but the references that have been pointed out in the bios theory page fail to give a mathematical definition. Sabelli is not mentioned in Mathematical Review, so I have not had much luck with that route. I have no idea what it means for dynamical system to display novelty: what properties must it satisfy? XaosBits 02:03, 16 January 2006 (UTC)

There is definition of novelty on the bios theory page, but it is unimportant for my question. Let me reformulate my question then: Do you personally really think that chaos (which is between the bifurcation cascade and the transition point at the x-axis value of around 4.6) and the pattern to the right of the transition point on the image above are the same thing? You don't really need to know anything about bios to answer this question.Lakinekaki 02:29, 16 January 2006 (UTC)

Yes, I do, but only because I've seen exactly this pattern before; this particular graph is not new news to people working in this field. There's even a name associated to the value to the right of the transition point at 4.6: its called the "winding number", and it counts how far you've moved as a function of the number of iterations. (The winding number for the region to the left of 4.6 is identically zero, since there is no movement.). Kolmogorov defined the winding number for this map (I believe) in the 1960's, maybe later, although Feigenbaum's const 4.6 was not yet known at that time, nor was the so-called "period-doubling route to chaos" understood. In the multi-colored graph I posted, the black regions are where the winding number is (effectively) zero (a const independent of the number of iterations, actually). The speckled, multi-colored regions are exactly the parts that are to the "right of 4.6" in your graph. They're speckled precisely because of the diffuse, wandering nature of the thing.

I'm finding this conversation tedious. I thought I'd made some reasonable statements, and instead of finding reasonable rebuttals, I'm finding lots of histrionics. linas 06:33, 16 January 2006 (UTC)

The type of motion plotted in the figure above is a short trajectory for x of the climbing sine map x + g sin(x), a map defined from the real line to itself. For small values of the control parameter g the motion is confined to cells that range from 2π(n-1) to 2πn, n integer. For larger values of g the motion can range over the entire real line. This happens when the first positive minimum of the map crosses the real axis, at g_d = 4.60333884875170035... (and not 4.604 as stated in some papers).

The image shown in this page is not a typical bifurcation diagram because it does not represent the range of values attained for the map for a particular value of g. Instead they are drawn “adiabatically”, with the value of g being incremented while keeping the last x as the new starting x. Numerical precision makes it difficult to observe a single trajectory ranging over the entire line. For values of g < g_d, the a portion of the image needs to be repeated up and down the y axis. For values of g > g_d the image would be a solid line, although of varying density.

For values of g > g_d (indicated biotic in the figure) the typical motion is to bounce inside one of the cells before hoping to a neighboring cell. The non-hyperbolic nature of the climbing sine map makes the long term behavior of this cell hoping subtle. It can be ballistic, diffusive, or anomalous diffusive. The diffusion coefficient for twice the distance-squared will be a fractal function of g. When the motion is diffusive, I would expect the dynamics to be chaotic.

A good starting point into the extensive literature of deterministic diffusion is the book of Pierre Gaspard, Chaos, Scattering and Statistical Mechanics or the upcoming book of Rainer Klages, Microscopic Chaos, Fractals, and Transport in Nonequilibrium Statistical Mechanics.

Are you suggesting me to read this book, or shall I send it to the authors of bios theory papers. I think that you are being arrogant and/or ignorant by telling me about the starting point book that deals with diffusion. What characterizes bios is not diffusion but the generation of novelty, which is absent in non-diffusive chaos but is present in non-diffusive bios! (it is independent from diffusion) Lakinekaki 06:54, 17 January 2006 (UTC)

N.B. The current definition of novelty in the bios theory page must be wrong. The number of isometries as defined is invariant under shuffling. I imagine something else was meant.

XaosBits 07:10, 16 January 2006 (UTC)

Summary as I perceive it

Your arguments fall into two categories:

1) In the first category I put arguments where you object that bios and biotic motion are not well defined (mathematically) on the Bios theory page, and since I put there something that is relatively new, it is upon me to provide enough information from the references cited in Bios theory article. You provide some good arguments there (single indent) and I give my comments (double indent):

I can't see how "biotic motion" is anything more than a special type of standard chaotic behavior. How exactly can you tell that a system is behaving "biotically" and not just chaotically (in the mathematical sense of the word)?
N.B. The current definition of novelty in the bios theory page must be wrong. The number of isometries as defined is invariant under shuffling. I imagine something else was meant.

I acknowledge this, and I have improved descriptions in Bios theory.

Maybe Bios is different, but the leading example provided -- An + 1 = An + gsin(An) is the "famous" circle map, studied for some 50-odd years now. To suddenly call this "biotic" is unusual. At+1 = At + g sin(At). Is known as the circle map.
Biotic motion seems to be a slice through the Standard Map, with some of the parameters varied adiabatically over time. Since the standard map is both well-known, heavily studied, and still full of many deep mysteries and surprises, it would be much more comforting if this example of biotic motion was compared to what is already known about the standard map, so that we could tell apart the new results from the parts that are "well known".
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.

What is new is not the circle map, but the new features discovered in it’s specific case, and also in other processes. Authors considered these features fundamental enough to define a new concept and publish papers on the subject. This does not mean that these processes cannot be analyzed with the tools devised for chaos analysis, on the contrary. (In the same manner that periodic series analysis is being applied to the chaotic processes) What this means is that there are additional characteristics in these processes that are not present in every process that can be put in the chaos category. I understand that you are making an argument that bios is only a subclass of chaos. However, in the same manner that periodicities have repeating values, while chaos series always generate variations, chaos always generates recurrences (absence of novelty), while bios always generates novelty. Bios is not subcategory of chaos, but a distinct concept. (James A. Yorke coined word chaos for processes analyzed for a long time. Recently word bios was given to another class of process that has been also analyzed for a long time and was mixed with chaos.)

For biotic motion there is no characterization of the type-of-motion, and consequently, no collection of results.

This is simply not true.

It is just that in a mature field, such as dynamical systems, the onus is on the discoverer to explain how the work fits in with the rest of knowledge.

I am not discoverer, but authors do explain this. Some explanations on Bios theory page seem quite clear to me. If you think it is not enough you can add more (this is Wikipedia!).

2) In the second category I put arguments where you object to the actual validity of things presented in papers (or the reputability of the journals), and to this argument I responded by trying to explain from my point of view why bios theory is valid. On this we can agree or disagree, but that is only our personal thing. Guidelines of Wikipedia are very clear in this regard. However, for the sake of clarifying things, I will provide summary of my comments here.

I thought I'd made some reasonable statements, and instead of finding reasonable rebuttals, I'm finding lots of histrionics.

You did make some reasonable statements, although I don’t see how they relate to the theory of bios (they relate to some equations, but not to the theory). Furthermore, did your statements disproof the theory of bios. If they did, how should that affect articles in Wikipedia? If they did not, what should I refute? Furthermore, I think I did give some reasonable responses.

Kolmogorov defined the winding number for this map (I believe) in the 1960's, maybe later, although Feigenbaum's const 4.6 was not yet known at that time, nor was the so-called "period-doubling route to chaos" understood.

Nor was so-called biotic motion known.

When the motion is diffusive, I would expect the dynamics to be chaotic.

Bios can be diffusive although it is not it’s fundamental characteristic.

I can see that all of you understood very well the kinetics of the process equation, and I am very glad for that. You mention what would happen when initial conditions originate in different basins of attraction, and I am providing here an image of that case. Six different basins generate this. Although you think that nothing fundamental happens after the point 4.6.. I tend to think that there is significant change happening there.

I also provide here three more graphs. In your view, left and middle images are different processes which they really are (left is periodicity with many points, while middle is chaos), while middle and right are the same (right is chaos). In my view (and the theory of bios), middle and right are also different processes (right is bios).

I don’t think that I have anything more to say on this subject, and since I think that the line that I am adding on the Chaos theory page about biotic motion is in full observance of the Wikipedia policies, I will keep adding this line. You can keep deleting it, but that will only lead to disputes and revert wars.

Verifiability does not mean that editors are expected to verify whether contents of an article are true. In fact, editors are strongly discouraged from conducting this kind of research.

Lakinekaki 06:54, 17 January 2006 (UTC)

(Some researchers distinguish a sixth type of behaviour, biotic motion, which they claim is distinct from chaotic motion.) This line seems very reasonable to me.Lakinekaki 16:31, 17 January 2006 (UTC)

biotic motion line

(In print, nine researchers distinguish a sixth type of behaviour, biotic motion, which they claim is distinct from chaotic motion.)

Maybe I have put references of that many, but there may be more papers on the bios theory that we may not be aware of, and therefore I think that the last solution was better:

(Some researchers distinguish a sixth type of behaviour, biotic motion, which they claim is distinct from chaotic motion.)

Lakinekaki 19:43, 17 January 2006 (UTC)

I decided to check the scope of publications covering biotic motion. Using two standard databases I was only able to find 8 publications in the period from 1997 to 2006. All publications are co-authored by Hector Sabelli. The two databases I used were the Mathematical Reviews and the Web of Science. I choose 1997 as the starting date because all references mentioned in the bios theory article are from 1998 or later.

None of the reviews in Mathematical Reviews mention either biotic motion or Sabelli's work.

A search in the Web of Science for the citations of Sabelli on biotic motion or the process equation from 1997 to 2006 reveals 8 publications, listed below, all of them co-authored by Sabelli. None of the work written since 1998 by one of the main proponents of biotic motion has been cited (outside of self-citations).

A search for process equation or biotic motion anywhere in the records of the articles covered by the Web of Science from 1997 to 2006, returns 142 hits of which 8 have anything to do with biotic motion. The publications are those already listed and co-authored by Sabelli.

In view of this search, I have modified the line in the main article. (Although removal would be more appropriate.)

XaosBits 23:26, 17 January 2006 (UTC)

I think that you should also include word bios, as the line refers to biotic motion while biotic motion is part of the bios theory. Authors acknowledging bios would implicitly acknowledge biotic motion. Till then, I reverted your change.Lakinekaki 23:45, 17 January 2006 (UTC)

There are zero articles in the Web of Science mentioning bios theory or bios in the sense of bios theory. So I'll revert the reverted line. XaosBits 02:50, 18 January 2006 (UTC)

I am impressed! You really did a good research. Now all you need to do is publish it somewhere (i.e. analysis, methodology used and results - and hopefully you will give me some credit there for telling you how to improve your results, and make your research more sound) so that it becomes verifiable since as we all know here, Wikipedia publishes things that are verifiable! Till then, I'll revert the change. Lakinekaki 03:31, 18 January 2006 (UTC)