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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm151, 16 pages, doi:10.1093/imrn/rnm151 published on January 8, 2008
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The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Homoclinic Orbits and Lagrangian Embeddings

Samuel T. Lisi

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, {omega} = d{lambda}) 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 {lambda}(XH) – H is positive and proper, H is positive outside a compact set and proper, and (W, {omega}) 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.



References

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
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Google Scholar
Right arrow Articles by Lisi, S. T.
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