The Equivariant Spectral Flow and Bifurcation of Periodic Solutions of Hamiltonian Systems

TL;DR

This paper introduces a $G$-equivariant spectral flow for symmetric Hamiltonian systems, providing a new tool to detect bifurcations of periodic solutions that may be missed by classical methods.

Contribution

It defines a novel $G$-equivariant spectral flow and applies it to analyze bifurcations in symmetric Hamiltonian systems, extending classical spectral flow concepts.

Findings

The $G$-equivariant spectral flow can be non-trivial even when classical spectral flow is zero.

It effectively detects bifurcations of periodic solutions in symmetric Hamiltonian systems.

The approach generalizes spectral flow to incorporate group symmetries.

Abstract

We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This $G$-equivariant spectral flow shares all common properties of the integer valued classical spectral flow, and it can be non-trivial even if the classical spectral flow vanishes. Our main theorem uses the $G$-equivariant spectral flow to study bifurcation of periodic solutions for autonomous Hamiltonian systems with symmetries.