On the heteroclinic connection problem for multi-well gradient systems

We revisit the existence problem of heteroclinic connections in RNRN associated with Hamiltonian systems involving potentials W:RN→RW:RN→R having several global minima. Under very mild assumptions on W   we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor W. Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of [22] and represents a more geometric alternative to the approaches of e.g. [5], [10], [14] and [17] for finding such connections.