Abstract
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 orbitūand we establish the existence of a heteroclinic orbit from 0 toū. This connecting orbit is obtained as the limit in theC1loctopology of a sequence of homoclinics with a winding number larger and larger.
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