Spectral flow, crossing forms and homoclinics of Hamiltonian systems

TL;DR

This paper establishes a spectral flow formula linking spectral flow and Maslov index in Hamiltonian systems, providing conditions for bifurcation of homoclinic trajectories in nonautonomous cases.

Contribution

It introduces a spectral flow formula relating spectral flow to the Maslov index for Hamiltonian systems with homoclinic boundary conditions, and derives bifurcation criteria.

Findings

Spectral flow formula connecting spectral flow and Maslov index.

Sufficient conditions for bifurcation of homoclinic trajectories.

Application to nonautonomous Hamiltonian vector fields.

Abstract

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of nonautonomous Hamiltonian vector fields.