Homoclinic solutions for eventually autonomous high-dimensional Hamiltonian systems

Holmes and Stuart [Z. Angew. Math. Phys. 43 (1992) 598–625] have investigated homoclinic solutions for eventually autonomous planar flows by analysing the geometry of the stable and unstable manifolds. We extend their discussion to higher-dimensional systems of Hamiltonian type by formulating the problem as the existence of intersection points of two Lagrangian manifolds. Their various assumptions can be restated and interpreted as ensuring some complexity of the generating function of one of the Lagrangian manifold with respect to symplectic coordinates that trivialise the second Lagrangian manifold. The critical points thus obtained correspond to homoclinic solutions. The main new feature in high-dimensions is that twice as many homoclinic solutions are found as for planar flows, in analogy with results obtained for autonomous Lagrangian systems by Ambrosetti and Coti Zelati [Rend. Sem. Mat. Univ. Padova 89 (1993) 177–194].  2002 Elsevier Science (USA). All rights reserved.