Abstract
We consider the numerical aspects of a Hopf bifurcation which occurs on a branch of travelling wave solutions in equations with O(2) symmetry. The Jacobian at every travelling wave solution is singular due to the O(2) symmetry which precludes the use of standard Hopf theory. Our approach is to add a spatial phase condition which removes the degeneracy in the Jacobian and allows standard theory to be applied. The numerical implications of this approach are also considered. The methods are applied to the Kuramoto-Sivashinsky equation. Numerical results are obtained which confirm a conjecture by Kevrekidis, Nicolaenko and Scovel (SIAM J. Appl. Math. 50, 760–790) and that are in agreement with results obtained by Armbruster, Guckenheimer and Holmes (SIAM J. Appl. Math. 49, 676–691) based on a centre-unstable manifold reduction which is not formally valid.
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References
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© 1992 Birkhäuser Verlag Basel
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Aston, P., Spence, A., Wu, W. (1992). Numerical Investigation of the Bifurcation from Travelling Waves to Modulated Travelling Waves. In: Allgower, E.L., Böhmer, K., Golubitsky, M. (eds) Bifurcation and Symmetry. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 104. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7536-3_4
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DOI: https://doi.org/10.1007/978-3-0348-7536-3_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-7538-7
Online ISBN: 978-3-0348-7536-3
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