Hopf bifurcation on a simple cubic lattice

We study Hopf bifurcation for differential equations defined on the space of functions on R^3 which are triply periodic with respect to a simple (primitive) cubic lattice. The centre manifold theorem reduces the problem to a system of ordinary differential equations (ODEs) on the space (C + C)^3 and symmetric under the group We abstract this group as the wreath product group and we use a general theory of symmetry-breaking bifurcations for wreath product groups to find (up to conjugacy) all branches of periodic solutions with maximal isotropy. The stability of these solutions is calculated. Branches of periodic solutions with sub-maximal isotropy can also exist. Some possibilities for bifurcations to heteroclinic cycles are explored.