The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems

TL;DR

This paper introduces a non-trivial index bundle for families of selfadjoint Fredholm operators on real Hilbert spaces and applies this to analyze multiparameter bifurcation phenomena in Hamiltonian systems.

Contribution

It demonstrates the existence of a non-trivial index bundle in real Hilbert spaces and extends bifurcation analysis of Hamiltonian systems using this new insight.

Findings

Existence of non-trivial index bundle for families parametrized by $X imes S^1$

Application of the index bundle to multiparameter bifurcation of homoclinic solutions

Generalization of previously known bifurcation examples

Abstract

The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and J\"anich in the sixties, and it is readily seen that on complex Hilbert spaces this so called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that for every real Hilbert space and every compact topological space $X$ there is a family of selfadjoint Fredholm operators parametrised by $X\times S^1$ which has a non-trivial index bundle. Further, we use this observation and a family index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.