We consider the Klein-Gordon equation on a Riemannian surface which is
globally well-posed in the energy space. This equation has an homoclinic orbit
to the origin, and in this paper we study the dynamics close to it. Using a
strategy from Groves-Schneider, we show that there are many solutions which
stay close to this homocline during all times. We point out that the solutions
we construct are not small.Comment: To appear in DCDS-A. The last part has been removed, and gives a
separate publicatio
