Hopf-Hopf mode interactions with O(2) symmetry

Abstract

In this paper we study the unfoldings of 0(2)-equivariant vector fields whose linearization has two pairs of purely imaginary eigenvalues. Such singularities may be expected to occur at isolated points in a centre manifold reduction of two-parameter systems with full circular symmetry. This situation differs from the corresponding non-symmetric system in that generically the eigenvalues may be either simple or double. The case when both eigenvalues are simple is similar to the Takens codimension-two singularity. Our interest lies in the cases where one or both of the purely imaginary eigenvalues are double; these cases lead to 6- and 8-dimensional centre manifolds, respectively. We use isotropy subgroup techniques to classify the types of solutions which occur. These include periodic solutions and 2-, 3-, and 4-dimensional invariant tori.