On the bifurcations of equilibria corresponding to double eigenvalues

Systems of ordinary differential equations having a finite symmetry group a re considered. One-parameter local bifurcations of symmetric equilibria cor responding to a double pair of purely imaginary eigenvalues are studied. It is shown that in one case a two-dimensional torus is generated from the equilibrium. The torus contains limit cycles; their number does not depend on the values of the parameter. The trajectories of the system that do not leave a certain fixed domain may only tend to the equilibrium under study o r to the 2-dimensional torus or to one of two (disjoint) limit cycles. In all the other cases an invariant surface is generated from the equilibri um which is diffeomorphic to the three-dimensional sphere. The behaviour of the trajectories on this surface depends on the symmetry group and is not studied in this paper. In the appendix we provide information on codimension 1 bifurcations corres ponding to double zero eigenvalues.