Talk:Chaos theory

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Contradiction: which is weaker?

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"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)[reply]

Both of those say transitivity is the weaker version. Sesquilinear (talk) 05:37, 17 April 2025 (UTC)[reply]

Chaos theory should be labeled as pseudoscience

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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)[reply]

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

Addition of a Section regarding the prediction of the behavior of chaotic systems.

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Hi all, I would suggest adding a new short section which explains the approaches to prevision of extreme events in chaotic systems. This could be a sub-section of the Section 'Applications'. I have taken these topics from a scientific article and, therefore, I would like to further modify the text before publishing it on the main page. I invite all interested Wikipedians to help me improve this paragraph, which I think is important for this topic.


Deterministic against probabilistic forecasting

According to the self-organized criticality model, chaotic systems produce events exhibiting intensities distributed according to a power law [citation], and extreme occurrences are considered inherently unpredictable in the sense that we cannot know, for example, whether the next price fluctuation of the Euro–US dollar exchange rate or the next earthquake will be ordinary or extreme. Nevertheless, an appropriate inferential analysis of the power-law probability distribution allows for long-term probabilistic forecasts of extreme events [citation]. Risk analysis in finance, seismology, or other fields, is based on the assumption of accepting a tolerable (small) probability that an extreme event will exceed a certain threshold.

For instance, in finance, knowledge of the distribution of price fluctuations, while not enabling short-term predictions, allows for risk analysis [citation]. When placing a buy or sell order on market stocks or other product, we cannot know whether the price variation will be favorable or not. Nevertheless, based on knowledge of the distribution of price fluctuations (in absolute value), we can estimate the probability that our loss may exceed a certain threshold within a specific time range, such as daily, weekly, annually, etc. MadameButterfly96 (talk) 12:30, 5 July 2025 (UTC)[reply]

But without a source I will oppose any addition. Johnjbarton (talk) 15:02, 5 July 2025 (UTC)[reply]
Dear @Johnjbarton, thank you for your reply. The text I proposed I took (and modified) from this article:
Power Law Distribution and Multi-Scale Analysis in Earth Sciences, Finance and other Fields: some Guidelines to Parameter Estimation, Chaos, 35, https://doi.org/10.1063/5.0259215
I was thinking of inserting a citation to this article in the points indicated with [citation]. However, I think the text should be modified a bit more. ~~~ MadameButterfly96 (talk) 15:25, 5 July 2025 (UTC)[reply]
Thanks for your reply and the citation. This article seems to be primarily a review of multi-scale analysis of power-law time series and a comparison of two methods. This isn't mainly about chaos theory, it is just one example. It's much more closely related to power law. The authors are not evidently experts in chaos theory or finance. The source has no citations. I think other sources should be included to verify the content. The first sentence above is not supported by the source. Johnjbarton (talk) 23:15, 5 July 2025 (UTC)[reply]
I agree with you that this paragraph does not address specific aspects of Chaos Theory, but rather focuses on certain statistical methods. The text you see was copied (with minor edits here and there to avoid plagiarism issues) from 'Section IV. Interpreting power-law distributions in inferential analysis' of the article I mentioned.
My idea was to summarize this section in a few lines to simply explain that deterministic prediction of chaotic systems is difficult, while probabilistic forecasting, when supported by proper statistical analysis, is reliable and has been used for decades in fields such as earthquake-engineering and risk analysis in economics and environmental studies.
I believe these concepts should be explained in this article. However, if you feel that these are off-topic, I’ll refrain from investing more time on them.
Let me know what you think. Thank you again for your suggestions. MadameButterfly96 (talk) 10:57, 6 July 2025 (UTC)[reply]
I think the goal you express in your paragraph "..explain that deterministic prediction.." is interesting. The example-driven source and description I also like. The part that I am less comfortable with is "..of chaotic systems..." because the source does not establish that it covers all chaotic systems. Nor does the source limit itself to chaotic systems. Rather the source covers power law based systems including some chaotic systems. To me a better approach would be to add a broader explanation to power law, forecasting, or prediction and a sentence linking that content in this article. eg "For chaotic systems with underlying power law distributions, like X, Y, and Z, probabilistic forecasting can be successful even if deterministic prediction fails.[citation]". Johnjbarton (talk) 17:57, 6 July 2025 (UTC)[reply]
Dear @Johnjbarton, sorry to bother you again with this article, and thanks so much for being willing to help.
Were you thinking of a draft similar to the one I am writing below?
Section proposal:
In many phenomena involving events exhibiting intensities distributed according to a power law, extreme occurrences can be forecasted in probabilistic rather than in deterministic terms, by means of appropriate statistical analyses [citation]. Risk analysis is based on the assumption of accepting a tolerable (small) probability that an extreme event will exceed a certain threshold. For chaotic systems with underlying power law distributions, such as earthquakes, forest fires, price fluctuations in stock markets, etc., probabilistic forecasting can be successful even if deterministic predictions fail.
By way of example, in finance, knowledge of the distribution of price fluctuations, while not enabling short-term predictions, allows for risk analysis. When placing a buy or sell order on market stocks or other financial products, we cannot know whether the price variation will be favorable or not. Nevertheless, based on knowledge of the probability distribution of price fluctuations (in absolute value), we can estimate the probability that our loss may exceed a certain threshold within a specific time range, such as daily, weekly, annually, etc. [citation] MadameButterfly96 (talk) 23:52, 7 July 2025 (UTC)[reply]
I encourage you to add this to Power law as a new section Power law § Forecasting, probably at the bottom of that page. I would make changes: add citation at the end of the first paragraph, drop "etc", reword to avoid "we", try be direct and compact, eg drop "By way" or even "By way of example". Once that is in place, add a sentence here that links this new section. Johnjbarton (talk) 02:49, 8 July 2025 (UTC)[reply]
One weakness with your proposed addition is the source is very new and does not have any citations. Other editors may challenge the source, even though I think it is much better than many web-site sources used on other pages. The source is a peer reviewed review in an AIP journal, but you can fortify the case by finding other sources that verify the key points. The source you have cites other work: look there for leads. Look those up on Google Scholar to find more. Review sources, textbooks, and source with >100 citations are the best. Johnjbarton (talk) 03:44, 9 July 2025 (UTC)[reply]
I see you both lack some math ... forget some math for now is not essential.
The power law is a critical exponent, this justify scaling laws and transition to chaos in critical systems, and there is some rationale about the RG. In the same manner there is the kolmogorov cascade which has a power law, i.e turbulence. Then single random events instead is stochastic kolmogorov jump processes with their own "stochastic" equations. These are just 3 different models for chaos. On top I would not touch sandpiles here cause is a chaotic attractor of a finite state automata, i.e this is too much rocket science for this article at this stage. I love the finance corner though which is it's own track, which I would fit in an "experimental" finance section in it's own glory. The article itself can be a good reference for a finance section, cause in essence is a review article, but as john says "too recent" no good. Try to stick to something like PRL / nature and famous authors as such.
I imagine you can easily edit a first draft of such a finance section, and keep it at "emergence of power laws in finance" .. but stay away from Landau...
e.g. this seems better[1] it has 1650 citations and the guy has an h-index of 60. and for example [2] At this point you can put the 3 of them (e.g. your one is data driven, second one goes along kolmogorov, third one goes with landau). Flyredeagle (talk) 19:21, 13 July 2025 (UTC)[reply]
Thank you so much, @Flyredeagle, for your suggestions. It would be really interesting to develop a section on experimental finance. Over the next few weeks, I'd like to try to gradually develop (always on this talk page) a paragraph on this topic. For now, I'll write a couple of lines as John suggested, mentioning that probabilistic forecasting analysis can be useful for some chaotic systems with underlying power law distributions. Thank you! MadameButterfly96 (talk) 17:03, 21 July 2025 (UTC)[reply]

References

  1. ^ https://pages.stern.nyu.edu/~xgabaix/papers/pl-ar.pdf
  2. ^ https://www.cfm.com/wp-content/uploads/2022/12/227-2000-power-laws-in-economics-and-finance-some-ideas-from-physics.pdf

Chaos in the History section.

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The history section is mess of duplicate and confusing claims. We need a good secondary source. Chaos: Making a New Science should be used as a source, not as a historical event in the history of chaos. Johnjbarton (talk) 20:54, 10 July 2025 (UTC)[reply]

I do agree that the history section is chaotic .. that's why I did not insisted in reverting my useful edit.
I also believe that having a logical flow is more important than just adding stuff in the article, i.e. less is more.
Now the chaos book is a good resource for post Lorenz Chaos.
For pre-post-lorenz for example there is: [1]
On top there is little mention of kolmogorov / arnold in general in the article which deserve a better place, e.g. turbulence is a form of chaos, same as bifurcations are a route to chaos and attractors are again a shape of chaos (e.g. sandpile).
Another interesting one seems [2] which mentions von neumann and birkhoff and the chaosbook [3] is a good free resource in any case.
I had a piece of section with some correct claims actually, that was in need of extra references and further expansion:
"Chaos theory began in the field of ergodic theory and from scientist like poincare. Planetery chaos happens on the scale of million of years, i.e. the Lyapunov exponents are very large, and therefore there is little experimental evidence for it, in the solar system there are still a few evidences of chaotic motion of non spherical objects such as chaotic rotation. At the end of the 19th century and concurrently to Weierstrass and Cantor,
Poincare research was also connected to early form of research on fractals such as the limit points of fuchsian groups, which are also a typical signature of chaos."
Actually the reference to long lyapunov exponents and Rotational chaos was to clarify a confusing statement that was there before about "non measurable" planetary chaos. (i.e. the dynamical system route to chaos). Given the long exponents are also mentioned above in the intro, it should be merged somehow.
The reference to fuchsian groups is actually correct, it was the starting point of poincare, to understand recursion of the linear fractional transformation, and from there automorphic functions, again these are relevant in the context of recursion, and are simple forms of fractals, and recursion links you logically to the chaos book i.e. Lorenz and mandelbrot.
(i.e. the fractal route to chaos)
Same apply for the weierstrass function and the cantor set, which are first examples of fractals, which appear again and again in the chaotic maps and elsewhere. Now these guys knew each other and were playing with some shape of "math of the devil".
I remember there was some historical book about the connection with chaos and end of 1800 math. Finally poincare needs some mention about poincare maps (with a nice pic such as a map of all possible orbits of a comet), and the way they plan space travel nowadays (again with a nice pic).
Plus one if you want to go fancy you can include fibonacci numbers, sunflowers, sphere packing and cauliflowers,
Phase transitions, turbulence transition, critical points and few others such as the hofstadter butterfly and hofstadter sequences (i.e. godel escher bach book).
Now all of these may not belong to a read through historical section, like in a GA article, but some / or part of it may belong more to a math corner (or expanded subpage/section), e.g. automophic functions and fuchsian groups (i.e. the hyperbolic geometry route to chaos) are essentially important because they are good didactic examples for automorphic forms, scaling and the modular group which are relevant in modern math and physics. (i.e. the number theoretic route to chaos and a p-adic thingy too).
Also the header section is too big, the introduction section is too small, the history section too big, not sure if it is best to have history at the beginning of the article or at the end....
I would also consider to start from something like the laplace daemon (e.g in the intro) and contextualize chaos from there.
Now first logical track in the article route(s) to chaos, second logical track recursion, third logical track is how to distinguish chaos (i.e. the dynamical systems section).
Other characteristics of chaos is a bit of a catch all that has his pros and cons.
If you can pick up some changes that would be nice, feel free to edit as you wish,
I prefer to review later and improve than agree upfront. The math side maybe fitting better in subpages, or in something like a separate page (mathematical formulations of chaos) and keep this article clean / GA style.
I saw you had a bit of discussion about deterministic/probabilistic above .. I will reply there.
Therefore yes .. lot of work to do...
Flyredeagle (talk) Flyredeagle (talk) 19:00, 13 July 2025 (UTC)[reply]

References

  1. ^ https://galileo-unbound.blog/2024/04/03/a-short-history-of-chaos-theory/
  2. ^ https://chaosbook.org/version17/chapters/appendHist-2p.pdf
  3. ^ https://chaosbook.org/chapters/ChaosBook.pdf