N. Filipski and M. Golubitsky
The abelian Hopf H mod K theorem
SIAM J. Appl. Dynam. Sys.
9 (2) (2010) 283-291.
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 Gamma. 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 Gamma is a finite
abelian group. We give examples of our results in the case when the symmetry
group Gamma = Z_l X Z_k acts on R^l X R^k 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.