# Viscosity solutions methods for converse KAM theory

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

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

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topGomes, 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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