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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.
@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 },
pages = {1047--1064},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
doi = {10.1051/m2an:2008035},
mrnumber = {2473319},
zbl = {1156.37015},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008035/}
}
TY - JOUR AU - Gomes, Diogo A. AU - Oberman, Adam TI - Viscosity solutions methods for converse KAM theory JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2008 SP - 1047 EP - 1064 VL - 42 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008035/ DO - 10.1051/m2an:2008035 LA - en ID - M2AN_2008__42_6_1047_0 ER -
%0 Journal Article %A Gomes, Diogo A. %A Oberman, Adam %T Viscosity solutions methods for converse KAM theory %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2008 %P 1047-1064 %V 42 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008035/ %R 10.1051/m2an:2008035 %G en %F M2AN_2008__42_6_1047_0
Gomes, Diogo A.; Oberman, Adam. Viscosity solutions methods for converse KAM theory. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 6, pp. 1047-1064. doi: 10.1051/m2an:2008035
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