Hopf Bifurcations on Generalized Rectangles with Neumann Boundary Conditions

Abstract

Bifurcation problems for PDEs posed on multidimensional rectangular sub-domains of Euclidean space often possess more symmetry than is immediately apparent. In particular, suppose that the partial differential operator is invariant under the subgroup of the Euclidean group generated by translations and by reflections in coordinate hyper-planes. Then solutions of the PDE may be extended periodically by reflecting them across the boundaries of the rectangle. These extra ‘hidden’ symmetries affect the generic bifurcation equations for mode interactions. In a previous paper we established the appropriate general forms of these bifurcation equations, for the interaction of two steady-state modes. Here we extend the analysis to interactions involving Hopf modes, namely a single Hopf mode, a steady-state/Hopf mode interaction, and a Hopf/Hopf mode interaction.