Viscosity solutions methods for converse KAM theory
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 6, p. 1047-1064

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.

DOI : https://doi.org/10.1051/m2an:2008035
Classification:  37J50,  49L25,  65P10,  70H7
Keywords: Aubry-Mather theory, Hamilton-Jacobi integrability, viscosity solutions
@article{M2AN_2008__42_6_1047_0,
author = {Gomes, Diogo A. and Oberman, Adam},
title = {Viscosity solutions methods for converse KAM theory},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
pages = {1047-1064},
doi = {10.1051/m2an:2008035},
zbl = {1156.37015},
mrnumber = {2473319},
language = {en},
url = {http://www.numdam.org/item/M2AN_2008__42_6_1047_0}
}

Gomes, Diogo A.; Oberman, Adam. Viscosity solutions methods for converse KAM theory. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 6, pp. 1047-1064. doi : 10.1051/m2an:2008035. http://www.numdam.org/item/M2AN_2008__42_6_1047_0/

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