Homoclinic Orbits and Lagrangian Embeddings
Mathematics Department, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA
Correspondence: Correspondence to be sent to: lisi{at}math.stanford.edu
This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Séré and to Coti-Zelati, Ekeland, and Séré [5]; [12], which were obtained by variational methods. Our existence result uses a modification of a construction due to Mohnke [10] (originally in the context of Legendrian chords), and an energy–capacity inequality of Chekanov [4]. In essence, we show the existence of a homoclinic orbit by showing a certain Lagrangian embedding cannot exist. We consider a (possibly time-dependent) Hamiltonian system on an exact symplectic manifold (W, 
= d
) with a hyperbolic rest point. In the case of periodic time dependence, we show the existence of an orbit homoclinic to the rest point if (XH) – H is positive and proper, H is positive outside a compact set and proper, and (W, ) admits the structure of a Weinstein domain. In the autonomous case, we establish the existence of an orbit homoclinic to the rest point if the critical level is of restricted contact-type, and the critical level has a Hamiltonian displaceable neighborhood.