HOPF-BIFURCATION WITH D3 X D3-SYMMETRY

A generic Hopf bifurcation involving an eight dimensional center eigenspace is considered for systems possessing a D3 x D3-symmetry. This kind of Hopf bifurcation can occur in systems of three interacting groups of oscillators, where each group itself is composed of three individual oscillators. The terminology ''micro'' and ''Macro'' is introduced here to denote symmetry operations acting on individual oscillators and on the whole groups, respectively. The normal form for the Hopf bifurcation admits 11 distinct periodic solutions with maximal isotropy subgroups. These are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The symmetry properties of these solutions when only certain Macro variables in the oscillator groups are observed are discussed in the context of the remaining symmetry. Furthermore, the relation of the normal form to the corresponding one for a single D3-symmetry is established by restricting the system to four dimensional fixed point subspaces associated with submaximal isotropy subgroups. Based on this information the possibility of quasiperiodic solutions and of a particular class of heteroclinic cycles is discussed.