Bifurcation of periodic solutions of ODEs with finite symmetry group

We are concerned with differential equations which are equivari- ant under the action of certain finite groups. In the first place we investigate the Hopf bifurcation and synchronization properties of a ZN × ZN −equivariant system of FitzHugh-Nagumo cells on a discrete torus; we also discuss certain oscillation patterns of two coupled tori. In the second place, we investigate which periodic solutions predicted by the H mod K theorem are obtainable by the Hopf bifurcation when the group is tetrahedral or octahedral. Finally, we investigate the num- ber of limit cycles that a family of Z6−equivariant polynomial planar systems can give rise to. These are results of my collaboration with Professors Mar ́ıa Jesu ́s A ́lvarez and Isabel S. Labouriau.