HOPF-BIFURCATION ON A SQUARE LATTICE

A complete classification of the generic D4 times sign with bar connected to left 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-equal 0 and frequency omega not-equal 0. The D4 times sign with bar connected to left 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 times sign with bar connected to left T2 x S1. On a neighbourhood of the Hopf bifurcation point, all small amplitude oscillatory solutions are found and their stabilities calculated relative to perturbations that preserve the spatial periodicity of the square lattice. There are five oscillatory solutions with maximal isotropy that bifurcate from the trivial solution; these are interpreted in terms of standing and travelling wave convection patterns. An unstable submaximal solution is also found to exist in an open region of parameter space. All possible bifurcation diagrams are given. The possibility of a primary bifurcation to a structurally stable heteroclinic cycle is explored.