The Cyclic Hopf H mod K Theorem

The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a $Γ-$equivariant dynamical system, based on the group-theoretical aspects. In addition, it classifies the spatio temporal symmetries that are possible. By the contrary, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each $\mathbf{C}-$axial subgroup of $Γ\times\mathbb{S}^1.$ In this paper we identify which periodic solution types, whose existence is guaranteed by the $H~\mathrm{mod}~K$ theorem, are obtainable by Hopf bifurcation, when the group $Γ$ is finite cyclic.

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