Talk:Hopf bifurcation

Diagram
A diagram showing a typical Hopf bifurcation would be useful. Particularly one that demonstrates the difference between subcritical and supercritical Hopf bifurcations. I suspect that there should be a diagram like that already in the public domain somewhere (if not online, then in an old textbook), but if not, maybe I'll come back and make one when I get some free time.

Dannya222 (talk) 06:08, 22 December 2007 (UTC)

I removed the link to Lyapunov exponent from Lyapunov coefficient. The exponent is quite different from the coefficient in the normal form of the Hopf. Ermentrout 17:40, 7 January 2006 (UTC)


 * I'm sure that's fine. By the way, we sign comments to talk pages, but we do not sign the articles themselves; the idea behind this is that we work together, and nobody "owns" an article (see Ownership of articles). Thanks for your work on the article, and welcome here. -- Jitse Niesen (talk) 22:04, 7 January 2006 (UTC)


 * I liked the part where it was all swirly. Penyulap   talk 10:23, 22 February 2012 (UTC)

Complex Variable Description
The use of complex variables in the introductory description introduces an unneed barrier to readers with less mathematical experience. The whole idea can be presented and described without having to make recourse to complex variables, atleast at the first-cut, by using explicit formulations and posing things in terms of rotational dynamics. —Preceding unsigned comment added by 146.186.131.40 (talk) 16:47, 4 October 2010 (UTC)

I agree. I also wonder if the complex normal form of a supercritical Hopf bifurcation couldn't be presented in a more digestible form, i.e.
 * $$ \dot{z} = z [ \lambda (r_0^2 - |z|^2) + i \omega_0 z ] $$

Then one sees immediately that $$ \lambda>0 $$ amounts to negative friction, rendering the quiescent state unstable, while the cubic nonlinearity stabilizes the amplitude to finite values. In fact the above form is a direct generalization of the Van-der-Pol oscillator. Benjamin.friedrich (talk) —Preceding undated comment added 21:26, 19 December 2011 (UTC).

opaque article
this article is incredibly opaque to anyone not already familiar with the subject. you can't define a hopf bifurcation as "a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane." without providing context for what a bifurcation even *is*.

2001:982:46CF:1:225:22FF:FEC6:F070 (talk) 04:06, 5 November 2015 (UTC)

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Question about the "definition"
The article currently state:
 * Suppose that all eigenvalues of J0 have negative real parts except one conjugate nonzero purely imaginary pair ±iβ. A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.

How can a "purely imaginary" pair of eigenvalues cross the imaginary axis? By definition, without real components, they are constrained to the imaginary axis, no? The only way they can "cross the imaginary axis" is if they share a real component that switches sign, right? -AndrewDressel (talk) 19:22, 15 October 2018 (UTC)

Utter lack of clarity
"In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises.[1] More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues - of the linearization around the fixed point - crosses the complex plane imaginary axis."

There is no doubt that the first two sentences could be written so that they are even harder to understand than they are now. But I cannot think of how this could happen.

By the unfortunate use of the word "bifurcation" before this word is defined here, this article obscures the very basic idea of this subject. It is entirely unmentioned that we are talking about a family of something that is parametrized by real numbers.

And by the unfortunate use of the word "system", the article doesn't even say that we are talking about dynamical systems, or vector fields, or differential equations.

I strongly believe that the article would be improved by the removal of this so-called introductory paragraph. But I won't do that.

Is there someone who is knowledgeable about the subject and also willing to make sure that what they write is clear and helpful? (Because if not, please find a different hobby.)2600:1700:E1C0:F340:B032:3AB1:3354:D41B (talk) 05:45, 26 October 2018 (UTC)

diagram labels
The following comments were directly in the article. I changed the diagram, since it followed a different sign convention. --W.pseudon (talk) 15:35, 1 July 2020 (UTC) In the figure it says first row alpha>0 and in the caption that supercritical are 1a) 1b) and 1c). In the text later it says alpha positive means subcritical. Which one is correct?   The labels on the figure are wrong. The top row should be alpha<0 and the bottom row should be alpha>0. Someone who has access to the original figure please correct this.

Applications
Hi, everybody. A new application of the Hopf bifurcation has been recently published in a JCR journal of Physics (https://doi.org/10.1088/1402-4896/abcad2) in connection with classical electrodynamics. I am the author of the paper, and therefore I am not the person allowed to upload the reference, since a COI would be at stake. But perhaps somebody could simply add two words (c classical electrodynamics \cite{}) after the Brusselator model. Sincerely Alvaro12Lopez. — Preceding undated comment added 07:33, 23 March 2021 (UTC)

Wrong reason
One sentence reads:

"Because 1 > 0 and −1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if 𝛍 = 0."

But this misstates the reason. A Hopf bifurcation does not occur or not occur because something is or is not obvious.

I hope someone knowledgeable about the Hopf bifurcation can fix this.

Is it a Hopf bifurcation ?
Soul surfer (talk) 10:31, 22 June 2023 (UTC)