B. Buffoni
Département de mathématiques,
Ecole Polytechnique Fédérale,
1015 Lausanne, Suisse
Boris.Buffoni@epfl.ch
Supported by a grant of the Swiss National Science
Foundation.
Date: May 2000
Holmes and Stuart [HS92] have investigated homoclinic orbits 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. The various assumptions introduced in [HS92] 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 orbits. The main new feature in high-dimensions is that twice as many homoclinic orbits are found as for planar flows, in analogy with results obtained for autonomous Lagrangian systems by A. Ambrosetti and V. Coti Zelati [ACZ93].
References
[ACZ93]
A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class
of conservative systems , Rend. Sem. Mat. Univ. Padova 89 (1993),
177-194.
[HS92]
P. J. Holmes and C. A. Stuart, Homoclinic orbits for eventually
autonomous planar flows , Z. angew. Math. Phys. 43 (1992), 598-625.