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The abelian Hopf H mod K theorem. (English) Zbl 1198.37088
The symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries are investigated. Main result is the abelian Hopf HmodK theorem which gives necessary and sufficient conditions for when these HmodK periodic solutions can occur by Hopf bifurcation when Γ is a finite Abelian group. Examples of these results in the case when the symmetry group . Γ= l × k acts on l × k by permutation of coordinates are given. In this case the HmodK periodic solutions that are obtainable by a generic Hopf bifurcation are classified and it is shown that there exist families of HmodK periodic solutions that cannot be obtained by Hopf bifurcation.
MSC:
37G40Symmetries, equivariant bifurcation theory
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE