Homoclinics and heteroclinics for a class of conservative singular Hamiltonian systems

We consider an autonomous Hamiltonian system $\ddot u+\nabla V(u)=0$ where the potential $V\colon\mathbb{R}^2\setminus\{\xi\}\to\mathbb{R}$ has a strict global maximum at the origin and a singularity at some point $\xi\ne 0$. Under some compactness conditions on $V$ at infinity and around the singularity $\xi$ we study the existence of homoclinic orbits to 0 winding around $\xi$. We use a sufficient, and in some sense necessary, geometrical condition $(*)$ on $V$ to prove the existence of infinitely many homoclinics, each one being characterized by a distinct winding number around $\xi$. Moreover, under the condition $(*)$ there exists a minimal non contractible periodic orbit $\bar u$ and we establish the existence of a heteroclinic orbit from 0 to $\bar u$. This connecting orbit is obtained as the limit in the $C^1_{{\rm loc}}$ topology of a sequence of homoclinics with a winding number larger and larger.