Abstract: Index theory revealed its prominent role in the study of periodic orbits of Hamiltonian systems and its dynamical consequences 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 and, as far as we know, no results at all, in the case of heteroclinics and halfclinics (i.e.parametrised by a half-line) orbits. Motivated by the importance played by these motions, we develop a new index theory and we prove at once a general spectral formula for heteroclinics, homoclinics and halfclinic trajectories. We will introduce the applications in N-body problem, FitzHugh-Nagumo systems and Bifurcations theory. This lecture is based on joint works with ChaoNien Chen, Alessandro Portaluri and Yuwei Ou.
