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Viscosity solutions methods for converse KAM theory

Published online by Cambridge University Press:  25 September 2008

Diogo A. Gomes  and
Adam Oberman
Diogo A. Gomes
Affiliation:
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. dgomes@math.ist.utl.pt
Adam Oberman
Affiliation:
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. dgomes@math.ist.utl.pt
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Abstract

The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, and the double pendulum.

Keywords

Aubry-Mather theoryHamilton-Jacobi integrabilityviscosity solutions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 1047 - 1064
Copyright
© EDP Sciences, SMAI, 2008

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