Hopf bifurcation on a simple cubic lattice

We study Hopf bifurcation for diff erential equations defined on the space of functions on R3 which are triply periodic with respect to a simple (primitive) cubic lattice. The centre manifold theorem reduces the problem to a system of ordinary diff erential equations (ODEs) on the space (C+C)3 and symmetric under the group (O=Zc2) + T3. We abstract this group as the wreath product group O(2) /S3, 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.