Homoclinics and Heteroclinics for a Class of Conservative Singular Hamiltonian Systems

We consider an autonomous Hamiltonian systemü+ V(u)=0 where the potentialV : R2\{ξ}→Rhas a strict global maximum at the origin and a singularity at some pointξ≠0. Under some compactness conditions onVat infinity and around the singularityξwe study the existence of homoclinic orbits to 0 winding aroundξ. We use a sufficient, and in some sense necessary, geometrical condition (*) onVto prove the existence of infinitely many homoclinics, each one being characterized by a distinct winding number aroundξ. Moreover, under the condition (*) there exists a minimal non contractible periodic orbit and we establish the existence of a heteroclinic orbit from 0 to . This connecting orbit is obtained as the limit in theC1loctopology of a sequence of homoclinics with a winding number larger and larger.