Viscosity solutions methods for converse KAM theory
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. firstname.lastname@example.org
Revised: 23 March 2008
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.
Mathematics Subject Classification: 37J50 / 49L25 / 65P10 / 70H7
Key words: Aubry-Mather theory / Hamilton-Jacobi integrability / viscosity solutions.
© EDP Sciences, SMAI, 2008