General bifurcations from periodic solutions with spatiotemporal symmetry, including mode interactions and resonances.

Jeroen Lamb, Ian Melbourne, Claudia Wulff.

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 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 vectorfields. 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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