| Issue |
ESAIM: M2AN
Volume 42, Number 6, November-December 2008
|
|
|---|---|---|
| Page(s) | 1047 - 1064 | |
| DOI | https://doi.org/10.1051/m2an:2008035 | |
| Published online | 25 September 2008 | |
Viscosity solutions methods for converse KAM theory
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. dgomes@math.ist.utl.pt
Received:
26
December
2006
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
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