Hopf bifurcation on a square lattice

Abstract

A complete classification of the generic D4*T2-equivariant Hopf bifurcation problems is presented. This bifurcation arises naturally in the study of extended systems, invariant under the Euclidean group E(2), when a spatially uniform quiescent state loses stability to waves of wavenumber k not=0 and frequency omega not=0. The D4*T2 symmetry group applies when periodic boundary conditions are imposed in two orthogonal horizontal directions. The centre manifold theorem allows a reduction of the infinite dimensional problem to a bifurcation problem on C4. In normal form, the vector field on C4 commutes with an S1 symmetry, which is interpreted as a time translation symmetry. The spatial and spatio-temporal symmetries of all possible solutions are classified in terms of isotropy subgroups of D4*T2*S1.