Geodesic
Viscosity solutions methods for converse KAM theory
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 6, pp. 1047-1064 Cet article a éte moissonné depuis la source Numdam

Voir la notice de l'article

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 : 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 },
     pages = {1047--1064},
     year = {2008},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {6},
     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

Cité par Sources :