Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions |
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Authors: | J.S.W. Lamb |
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Affiliation: | a Department of Mathematics, Imperial College, London SW7 2BZ, UK b Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK c Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK |
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Abstract: | We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields.We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework. |
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Keywords: | 37G40 37G15 37C55 |
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