Viscosity solutions methods for converse KAM theory

Diogo A. Gomes; Adam Oberman

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 6, page 1047-1064
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Gomes, Diogo A., and Oberman, Adam. "Viscosity solutions methods for converse KAM theory." ESAIM: Mathematical Modelling and Numerical Analysis 42.6 (2008): 1047-1064. <http://eudml.org/doc/250359>.

@article{Gomes2008,
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. },
author = {Gomes, Diogo A., Oberman, Adam},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Aubry-Mather theory; Hamilton-Jacobi integrability; viscosity solutions.; viscosity solutions},
language = {eng},
month = {9},
number = {6},
pages = {1047-1064},
publisher = {EDP Sciences},
title = {Viscosity solutions methods for converse KAM theory},
url = {http://eudml.org/doc/250359},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Gomes, Diogo A.
AU - Oberman, Adam
TI - Viscosity solutions methods for converse KAM theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/9//
PB - EDP Sciences
VL - 42
IS - 6
SP - 1047
EP - 1064
AB - 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.
LA - eng
KW - Aubry-Mather theory; Hamilton-Jacobi integrability; viscosity solutions.; viscosity solutions
UR - http://eudml.org/doc/250359
ER -

References

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