The equivariant spectral flow and bifurcation for functionals with symmetries: part I

Abstract

We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a recent construction of an equivariant spectral flow by the authors, we obtain a bifurcation theorem that generalises well-established results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and Recht. Finally, we discuss first examples of strongly indefinite systems of differential equations where the mentioned classical approaches fail but an invariance under an orthogonal action of a compact group makes our methods applicable and yields the existence of bifurcation.