The Abelian Hopf H mod K Theorem

We study the symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries. The H mod K Theorem gives necessary and sufficient conditions for the existence of periodic solutions with spatial symmetries K and spatio-temporal symmetries H in systems with finite symmetry group Γ. Our main result, the Abelian Hopf H mod K Theorem, gives necessary and sufficient conditions for when these H mod K periodic solutions can occur by Hopf bifurcation when Γ is a finite abelian group. We give examples of our results in the case when the symmetry group Γ = Zl×Zk acts on Rl × Rk by permutation of coordinates. In this case, we classify the H mod K periodic solutions that are obtainable by a generic Hopf bifurcation and show that there exist families of H mod K periodic solutions that cannot be obtained by Hopf bifurcation. AMS Subject Classification Codes 34C23, 34C25, 37G40