Wikipedia:Reference desk/Archives/Mathematics/2011 May 30

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May 30

Centre Manifold and Limit Cycle

Can somebody explain to me the idea of a stable manifold, centre manifold, and unstable manifold, in the context of a system with a stable limit cycle?

I have 2D a system with one parameter that I can vary, and at small values of the parameter the system has a stable equilibrium (a spiral), and when I raise the value of the parameter it undergoes a (Supercritical) Hopf bifurcation, with the equilibrium becoming unstable and all trajectories tending towards a stable limit cycle that orbits the equilibrium.

Where does the concept of 'manifolds' come into this? I'm not sure if the manifolds correspond to trajectories or what exactly they are. Does the centre manifold only exist at the point of bifurcation? 130.102.158.15 (talk) 05:54, 30 May 2011 (UTC)

We have articles on stable manifold, centre manifold and Hopf bifurcation. Don't let the term "manifold" confuse you - in your 2D system the stable manifold may be a single point or a closed curve i.e. a limit cycle. Higher dimensional manifolds (surfaces etc.) arise in systems with a phase space of three or more dimensions. Gandalf61 (talk) 08:07, 30 May 2011 (UTC)

Integration

Is it possible to integrate dx/x from -h to +h? If yes, how can we do it?203.199.205.25 (talk) 11:08, 30 May 2011 (UTC)

There's a discontinuity at x=0, so the integral is improper, and is usually interpreted as

$\int _{-h}^{h}{\frac {dx}{x}}=\lim _{b\to 0}\int _{-h}^{b}{\frac {dx}{x}}+\lim _{a\to 0}\int _{a}^{h}{\frac {dx}{x}}$

Neither of limits on the right exist, so the original integral does not exist. For a variation which does give an answer (it's 0), you might like Cauchy principal value. Staecker (talk) 11:28, 30 May 2011 (UTC)

One small point: we integrate 1 / x with respect to x, and that is written as

$\int {\frac {1}{x}}\,\operatorname {d} x=\int {\frac {\operatorname {d} x}{x}}\,.$

If you want to integrate dx / x then you have a double integral, and no idea what integral you want to integrate with respect to:

$\int \int {\frac {1}{x}}\,\operatorname {d} x\,\operatorname {d} ?=\int \int {\frac {\operatorname {d} x}{x}}\,\operatorname {d} ?\,.$

Maybe this is a technical convention, maybe other people say it another way. — Fly by Night (talk) 20:53, 30 May 2011 (UTC)

This is a pretty standard convention for naming the integral. It is the integral of the differential form dx/x. Sławomir Biały (talk) 21:20, 30 May 2011 (UTC)

Maybe, but if you want to integrate a differential form then you have to specify the base manifold. I get the feeling that the OP has never met differential forms, nor manifolds. Otherwise: why ask the question they asked? The usual, elementary language is "how to integrate ƒ(x) with respect to x". Just to support that point: please tell me how to integrate dx/x from −h to h on a manifold M. — Fly by Night (talk) 23:12, 30 May 2011 (UTC)

It's the integral of dx/x over the oriented manifold [-h,h]. Sławomir Biały (talk) 23:45, 30 May 2011 (UTC)

Thanks for telling me what my M was, and adding a boundary. Great answer :-) — Fly by Night (talk) 00:28, 31 May 2011 (UTC)