D4-symmetric Maps with Hidden Euclidean Symmetry

Abstract

Bifurcation problems in which periodic boundary conditions (PBC) or Neumann boundary conditions (NBC) are imposed often involve partial differential equations that have Euclidean symmetry. In this case posing the bifurcation problem with either PBC or NBC on a finite domain can lead to a symmetric bifurcation problem for which the manifest symmetries of the domain do not completely characterize the constraints due to symmetry on the bifurcation equations. Additional constraints due to the Euclidean symmetry of the equations can have a crucial influence on the structure of the bifurcation equations. An example is the bifurcation of standing waves on the surface of fluid layer. The Euclidean symmetry of an infinite fluid layer constrains the bifurcation of surface waves in a finite container with square cross section because the waves satisfying PBC or NBC can be shown to lie in certain finite-dimensional fixed point subspaces of the infinite-dimensional problem. These constraints are studied by analyzing the finite-dimensional vector fields obtained on these subspaces by restricting the bifurcation equations for the infinite layer. Particular emphasis is given to determining which bifurcations might reveal observable effects of the rotational symmetry of the infinite layer. It turns out that a necessary condition for this possibility to arise is that the subspace for PBC must carry a reducible representation of the normalizer subgroup acting on the subspace. This condition can be met in different ways in both codimension-one and codimension-two bifurcations.