NORMAL FORMS FOR DRIVEN SURFACE-WAVES - BOUNDARY-CONDITIONS, SYMMETRY, AND GENERICITY

The parametric excitation of standing waves in a vertically oscillated fluid can be studied by analyzing the period-doubling bifurcation in the associated stroboscopic map. When the fluid container has symmetry GAMMA the stroboscopic map will exhibit this symmetry and this in turn significantly affects the structure of the bifurcation problem. For both square and circular containers the normal forms appropriate to the reduced maps for the critical mode amplitudes are discussed. The experiments of Simonelli and Gollub with square containers strongly suggest that the effective symmetry of the problem is larger than the obvious geometric symmetry of the boundaries. For the ideal fluid model of Benjamin and Ursell with Neumann boundary conditions (NBC) one can show that the problem does indeed have a larger symmetry because it embeds in a model on a larger domain with periodic boundary conditions (PBC); the original NBC are then realized as a symmetry constraint on the solutions of this extended model. The additional symmetries introduced in this way lead to a large number of effects that would otherwise be non-generic including critical eigenspaces carrying reducible representations of GAMMA and reduced maps on these eigenspaces that have symmetries larger than GAMMA. These extra symmetries relate in-phase and out-of-phase mixed modes and lead to new branches to pure modes. The case of circular geometry is also discussed as it does not allow this extension to PBC and thus provides an instructive contrast with the subtleties found for square geometry.