# Index theory for heteroclinic orbits of Hamiltonian systems

## TL;DR

This paper develops a new index theory for heteroclinic, homoclinic, and halfclinic orbits in Hamiltonian systems, extending the well-established periodic case to these less understood trajectories and providing a spectral flow formula.

## Contribution

It introduces a novel index theory for heteroclinic and related orbits in Hamiltonian systems, filling a gap in the existing mathematical framework.

## Key findings

- Proved a general spectral flow formula for heteroclinic, homoclinic, and halfclinic trajectories.
- Revealed how the new index theory can recover classical results for orbits on bounded intervals.
- Extended the applicability of index theory beyond periodic orbits to more complex trajectories.

## Abstract

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrised by bounded intervals.