JOURNAL OF DIFFERENTIAL EQUATIONS 21,431-438 (1976) Homoclinic Orbits in Hamiltonian Systems ROBERT L. DEVANEY Department of Mathematics, Northwestern University, Evanston, Illinois 60201 ReceivedJanuary 25, 1975 The object of this paper is to study the orbit structure of a Hamiltonian system in a neighborhood of a trajectory which is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit. For diffeomorphisms, the analogous situation is fairly well understood. By a theorem of Smale [7], in every neighborhood of a transversal homoclinic point of a periodic point, there is a compact invariant set on which some iterate of the diffeomorphism is topologically conjugate to the Bernoulli shift on N symbols. Via Poincare maps on local transversal sections, there is thus an analogous result for hyperbolic closed orbits of vector fields. For critical points of vector fields, however, the situation is somewhat different. In the first place, by the Kupka-Smale theorem [4], the existence of orbits doubly asymptotic to an equilibrium point is not generic. In fact, the set of vector fields which admit homoclinic orbits at critical points is of the first Baire category in the set of all smooth vector fields. This follows since the sum of the dimensions of the stable and unstable manifolds at the critical point is at most equal to the dimension of the mani- fold itself. Hence, the stable and unstable manifolds cannot intersect trans- versely in a one-dimensional (homoclinic) orbit. For Hamiltonian systems, however, this is no longer true. The stable and unstable manifolds of hyperbolic critical points must both lie in a fixed energy surface, and by [lo], th e y are generically transverse within that surface. Since the codimension of energy surfaces is one, it follows that the stable and unstable mnaifolds may thus intersect transversely within the energy surface along a homoclinic orbit. Hence these orbits cannot be removed by small perturbations of the Hamiltonian. We remark that, with certain restrictions on the characteristic exponents at the critical point, Silnikov [5] has found horseshoe mappings similar to those of Smale near homoclinic orbits of vector fields. However, his assumptions 431 Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector