The Cyclic Hopf H mod K Theorem

TL;DR

This paper explores the relationship between the $H~\mathrm{mod}~K$ theorem and Hopf bifurcation in finite cyclic symmetry groups, identifying which periodic solutions can arise via Hopf bifurcation.

Contribution

It establishes criteria for which $H~\mathrm{mod}~K$ solutions are obtainable through Hopf bifurcation in cyclic groups.

Findings

Identifies which $H~\mathrm{mod}~K$ solutions are realizable by Hopf bifurcation.

Provides a classification of periodic solutions in cyclic symmetry contexts.

Bridges the gap between group-theoretic classification and bifurcation theory.

Abstract

The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a $\Gamma-$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 $\Gamma\times\mathbb{S}^1.$ 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 $\Gamma$ is finite cyclic.