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Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems

Holmes, Philip J. and Marsden, Jerrold E. (1982) Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems. Journal of Mathematical Physics, 23 (4). pp. 669-675. ISSN 0022-2488. http://resolver.caltech.edu/CaltechAUTHORS:HOLjmp82

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Abstract

We start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates. We use Kolmogorov–Arnold–Moser (KAM) theory to show that some of the resulting tori persist under small perturbations and use a vector of Melnikov integrals to show that, under suitable hypotheses, their stable and unstable manifolds intersect transversely. This transverse intersection is ultimately responsible for Arnold diffusion on each energy surface. The method is applied to a pendulum–oscillator system.


Item Type:Article
Additional Information:Copyright © 1982 American Institute of Physics Received June 17, 1981; accepted for publication 23 October 1981. Research partially supported by ARO contract DAAG-29-79-C-0086 and by NSF grants ENG 78-02891 and MCS-78-06718. We thank Alan Weinstein for several helpful discussions and Allan Kaufman for suggesting a stimulating physical example.
Subject Keywords:OSCILLATORS; PENDULUMS; COUPLING; HAMILTONIANS; ORBITS; INTEGRALS; TWO–DIMENSIONAL SYSTEMS; FORCES; DISTURBANCES; COORDINATES; DYNAMICS
Record Number:CaltechAUTHORS:HOLjmp82
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:HOLjmp82
Alternative URL:http://dx.doi.org/10.1063/1.525415
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4637
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:30 Aug 2006
Last Modified:26 Dec 2012 09:00

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