HOMOCLINICS AND HETEROCLINICS FOR A CLASS OF CONSERVATIVE SINGULAR HAMILTONIAN-SYSTEMS

We consider an autonomous Hamiltonian system <(u) over bar> + del(u)=0 where the potential V:R-2\{xi} --> R has a strict global maximum at t he origin and a singularity at some point xi = 0. Under some compactne ss 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 su fficient, and in some sense necessary, geometrical condition () on V to prove the existence of infinitely many homoclinics, each one bring characterized by a distinct winding number around xi. Moreover, under the condition () there exists a minimal non contractible periodic orb it <(u) over bar> and we establish the existence of a heteroclinic orb it from 0 to <(u) over bar>. This connecting orbit is obtained as the limit in the C-loc(1) topology of a sequence of homoclinics with a win ding number larger and larger. (C) 1997 Academic Press.