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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.
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 Wikipedia:Ownership of articles). Thanks for your work on the article, and welcome here. -- Jitse Niesen (talk) 22:04, 7 January 2006 (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 188.8.131.52 (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.
Then one sees immediately that 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).
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*.