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An algebraic criterion for symmetric Hopf bifurcation. (English) Zbl 0788.58038
From the authors’ abstract: “The equivariant Hopf bifurcation theorem states that bifurcating branches of periodic solutions with certain symmetry exist when the fixed-point subspace of the subgroup of symmetries is two-dimensional. We show that there is a group-theoretic restriction on the subgroup of symmetries in order for that subgroup to have a two-dimensional fixed-point subspace in any representation. We illustrate this technique for all irreducible representations of $SO\left(3\right)$ on the space ${V}_{\ell }$ of spherical harmonics for $\ell$ even”.
##### MSC:
 37G99 Local and nonlocal bifurcation theory 34C23 Bifurcation (ODE) 34C25 Periodic solutions of ODE 37C80 Symmetries, equivariant dynamical systems
##### Keywords:
Hopf bifurcation; periodic solutions; symmetry