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Hopf bifurcation and symmetry: travelling and standing waves on the circle

Published online by Cambridge University Press:  14 November 2011

Stephan A. van Gils  and
John Mallet-Paret
Stephan A. van Gils
Affiliation:
Department of Mathematics and Computer Science, Free University, P.O. Box 7161, 1007 MC Amsterdam, The Netherlands
John Mallet-Paret
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A.
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Synopsis

In this paper we consider Hopf bifurcation in the presence of O(2) symmetry. The system of reaction diffusion equations ut, = D(µ)uxx + f(µ, u) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 3-4 , 1986 , pp. 279 - 307
Copyright
Copyright © Royal Society of Edinburgh 1986

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