Talk:Bifurcation theory

Untitled

Wikipedia lacks entries on "Center manifold theorem" and "transformation to normal form" --Anonymous

Ok... ElectroChip123 (talk) 00:32, 25 March 2019 (UTC)

Lets make the article A-Class

Let us make a concerted effort to make this article A-Class as soon as we can. Here is list of topics working on which should lead to improvement in the quality of the article:

1. Making the language more rigorous: There are several statements in the article that are not mathematically precise. While such statements are make the article more readable at times, but I think that in this case many need to be re-written. Consider, for example, the first line of the article: "a bifurcation occurs when a small smooth change made to the parameter". This statement means that there can indeed be a non-smooth small change in a parameter, which is incorrect . Later in the article there are claims to the effect that all bifurcations are related to eigenvalues crossing the imaginary axis, which is not true (flip bifurcation or global bifurcations being counter examples). Also, not all global bifurcations occur due to "collision" of two objects in the phase-space. In short, there are many language and detail related changes that need to be made.
2. More "coverage" required: Presently the article does not cover many important topics in bifurcation theory. Here is an incomplete list of topics that need to be included:

• Theory of normal forms
• Center manifold theory
• Bifurcation diagrams
• Poincare maps
• Perturbation methods (near critical points)
• Method of multiple scales (near critical points)
• Bifurcation of DAEs and related phenomena
• Examples and diagrams for each kind of bifurcation
• Computational aspects of bifurcation theory
• Discussion of software tools like AUTO, MatCont, Content, BifPack, PyCont etc

NOTE: This section is under active development. Please contribute to it and the main article.

- - -

• Well, not to rain on anyone's parade, but -- alas -- the concept of "A-Class" presupposes a specific 'class' with respect to the audience's level of education & specialization. Already, even within the above request, Wikipedia mathematics articles in general slip almost unconsciously, almost always, into inaccessible jargon. This is not just an affliction of Wikipedia, by the way: MOST mathematicians can't access most math journal articles. Fact...and it is due to specialization within mathematics. If I could wave a magic wand and fix the problem (regrettably, I cannot) it would be to create a CAUTION label for *all* math articles on Wikipedia that reminds editors that all articles -- even mathematical -- can, should and must be written at the 7th grade reading level...just as "real" encyclopedias are. As regards opinions on this apparently intractable problem, "Your mileage may vary." Be well...and be honest (we all need you to be as such...and so do you). --104.15.130.191 (talk) 19:19, 14 June 2020 (UTC)

well I would argue that we need to have a tag for articles or sections

• 7th grade reading level - children encyclopedia
• 12th grade reading level - standard encyclopedia (may assume some e.g. calculus / literature etc. )
• Graduate level (i.e. bachelor) of the specific field - advanced articles (e.g. may assume knowledge about eigenvalues)
• All the rest

I don't see nothing wrong, that there are advanced topics in wikipedia or research trends but it shall be clearly stated, and please no obscure references (half thoughts, half research ideas without citations), please do these on the talk page. In practice there is a lot of research underground of people that ends up editing on wikipedia, and frustrated mathematicians, researchers or professionals that don't know what to make out of it (at least in a proper peer reviewed article you should always have a verified citation that you can dig through). The articles in the area of dynamical systems have also different expectations of "level of rigor" by the different audiences, due to both educational level and background, e.g. if I am a PHD student in biology I may have a problem understanding eigenvalues given is not my field, so it becomes double difficult.

The real problem is time: for most professional researcher editing wikipedia is typically unpaid volunteer work not contributing to the field (where actually good educational material is still foundational for good research). i.e easy to ask to do it, difficult to put time and effort to do it, and also to stay behind some odd wikipedians.

Flyredeagle (talk) 11:18, 21 December 2024 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Bifurcation theory/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Can be made more accessible. Add short descriptions of the bifurcations; normal form theory. Perhaps importance should be high. -- Jitse Niesen (talk) 12:30, 24 May 2007 (UTC)

Last edited at 12:30, 24 May 2007 (UTC). Substituted at 09:36, 29 April 2016 (UTC)

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Proposed Merger

The articles on Indefinite-period, homoclinic, and heteroclinic bifurcation are all stubs. The information in them can easily be fit into this article by expanding the Global bifurcations section. Furthermore, this merge will improve the quality this article by adding more information and more sources to it. ElectroChip123 (talk) 00:29, 25 March 2019 (UTC)

  Y Merger complete. Klbrain (talk) 07:31, 25 April 2020 (UTC)

Motif

I deleted this claim from the lede:

"Henri Poincaré also later named various types of stationary points and classified them with motif."

Motive (algebraic geometry) appears to offer an explanation of what a "motif" is. But it seems to be a branch of maths that was developed after Poincaré's time. I can't find anything that links Poincaré with motifs. The claim is in the lede, which is supposed to be a summary of content in the main body; but this claim isn't a summary of anything in the body.

I'm not a mathematician, and I can't understand what I've read about motives/motifs.

MrDemeanour (talk) 08:51, 26 July 2023 (UTC)

The claim is somewhat correct but every time you read algebraic geometry you need a dictionary of terms that changed over time (i.e. were generalized) to your own specific domain of expertise and you need a lot of other knowledge (such as abstract algebra and algebraic topology if not even number theory) to understand it properly.

• Poincare invented simplicial cohomology (1890s).
• At the time of poincare there was not even Volterra algebraic geometry (1910s), this was about special points, double points, points at infinity, asymptotes, non differentiable points, blow ups etc. Poincare had already a rather precise, although very concrete, idea of these.
• Groethendiek invented motives in the 1960s (which is a vast generalization of all cohomology theories), and all that goes with groethendiek is vastly abstract.
• Most applications of algebraic geometry to physics, such as dynamical systems, started after the end of 1970s
• The claim is at best a recent trend in research, if not even actual research.

You shall understand a motif as the cohomology structure (i.e. the "first order differentiability structure") of an algebraic variety (i.e. a set of "polynomial" equations). Note that these methaphors may horrify a "professional" and "pure" mathematician (e.g. "first order differentiability" is not first order cohomology group, and "polynomial" is very different from e.g. meromorphic), but birds are birds and frogs are frogs.

For poincare you started from a symplicial complex (i.e. a triangulation of the algebraic variety or more in general a manifold), you get to the dual, i.e the symplicial cohomology, and when the manifold is not smooth, vertixes of the triangualations collapse one onto the other and you get all these special points, according to how they collapse you get non differentiability in various ways (e.g. two tangents in one point).

Now you should see also that for poincare a set of polynomial equations, was not very far from a set of ordinary differential equations and not very far from a set of PDEs, therefore his idea was that there was one single theory encompassing all of these, ultimately they are all manifolds. For him classifying stability point in a smooth hamiltonian flow, classifying non differentiable points in a PDE or classifying special points in an algebraic variety was much of the same science.

Modern algebraic geometry instead is mostly about algebraic equations, not about about differential equations, this permits a large unification on the side of number theory but leaves differential equations as some shape of dark corner of applied mathematics. More recently (i.e. 2020s) this is changing but is quite a hard topic. Ultimately a special point is just one point and you would expect to use all tools available (i.e. from algebraic geometry and topology to differential equations and geometry) to classify it.

Just to throw a few examples in:

• the onsager anomaly is a non differentiable point in the boltzman equations when the viscosity goes to zero, that is actually one starting point for turbulence. Ultimately it is a shape of phase transition and ultimately is a very complex, not understood, form of bifurcation.
• The Higgs mechanism where by variation of the order parameter, from one stable point that becomes unstable, you get two extra stable points around it.
• Topological defects are essentially again various type of complex non differentiable points and they typically also have one or more parameters where the topology (ie. "connectivity" in this case) changes
• Chiral anomalies are again some shape of now complex special points

Now all of these examples are typically not linked together in a standard education curriculum, e.g this page is considered low priority in the physics portal, and dynamical system often is interpreted as "classical" system. They are interdisciplinary, hard topics and often they are not tractable problems, i.e. outside of the realm of current mathematics (i.e. math for the next century) or computation, but hard problems lead to new methods.

Now I left this as a kind of draft for a future re-addition in the page (definitely not in the lead). It needs to be polished up a bit cause it may not stand to the review of a "professional" and "pure" mathematician. Flyredeagle (talk) 10:27, 21 December 2024 (UTC)