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Bifurcation of periodic solutions of differential equations with finite symmetry groups

In this PhD thesis we are concerned with differential equations which are equivariant under the action of certain finite groups. The thesis consists of three articles. In the first place we investigate the Hopf bifurcation and synchronization properties of a N × N system of FitzHugh-Nagumo cells, the whole system of coupled cells being invariant under the ZN × ZN group. We are interested in the periodic solutions arising at a first Hopf bifurcation from the fully synchronized equilibrium. Our approach involves the coupling types between the cells in the array: associative couplings reduce the difference between consecutive cells, while for dissociative couplings differences are increased. In the associative type of coupling we find bifurcation into a stable periodic solution where all the cells are synchronized with identical behavior. When the coupling is dissociative in either one or both directions, the first Hopf bifurcation gives rise to rings of N fully synchronized cells. All the rings oscillate with the same period, with a 1 N -period phase shift between rings. When there is one direction of associative coupling, the synchrony rings are organized along it. Dissociative coupling in both directions yields rings organized along the diagonal. The stability of these periodic solutions was studied numerically and they were found to be unstable for small numbers of cells, stability starts to appear at N > 11. 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 Γ is tetrahedral and/or octahedral. We find that not all periodic solutions predicted by the H mod K theorem occur as primary Hopf bifurcations from the trivial equilibrium. We also analyze submaximal bifurcations taking place in 4−dimensional invariant subspaces and giving rise to periodic solutions with very small symmetry groups. In the third place, we investigate the number of limit cycles that a family of Z6−equivariant polynomial planar system can give rise to. This question is related to the Hilbert XVI problem. The goal is studying the global