Marek Izydorek

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.

THE LS-INDEX: A SURVEY

Let H be a Hilbert space. Consider a vector field f defined on H of the form f (x) = L(x) + K(x), where L is a strongly indefinite bounded linear operator and K is a completely continuous perturbation. In this paper we present an extension of the classical Conley index to flows generated by f as above. This topological invariant is called theLSindex. Furthermore, we describe an equivariant version of this index which generalizes the equivariant Conley index introduced by Floer. We also present cohomologicalLS-indices in both nonequivariant and equivariant cases. We discuss applications of these indices to existence and multiplicity results for periodic solutions of Hamiltonian systems. This survey is based on the author’s lectures delivered at the NATO-ASI – SMS 2004 at the Université of Montréal.