Viscosity solutions methods for converse KAM theory

Gomes, Diogo A.; Oberman, Adam

doi:10.1051/m2an:2008035

Gomes, Diogo A. ; Oberman, Adam

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 6, pp. 1047-1064.

Résumé

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.

MR Zbl

DOI : 10.1051/m2an:2008035

Classification : 37J50, 49L25, 65P10, 70H7
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     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://www.numdam.org/articles/10.1051/m2an:2008035/}
}

TY  - JOUR
AU  - Gomes, Diogo A.
AU  - Oberman, Adam
TI  - Viscosity solutions methods for converse KAM theory
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 1047
EP  - 1064
VL  - 42
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/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: Modélisation mathématique et analyse numérique
%D 2008
%P 1047-1064
%V 42
%N 6
%I EDP-Sciences
%U http://www.numdam.org/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: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 6, pp. 1047-1064. doi : 10.1051/m2an:2008035. http://www.numdam.org/articles/10.1051/m2an:2008035/

Bibliographie
Cité par

[1] V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical aspects of classical and celestial mechanics. Springer-Verlag, Berlin (1997). Translated from the 1985 Russian original by A. Iacob, reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems III, Encyclopaedia Math. Sci. 3, Springer, Berlin (1993) MR 95d:58043a]. | MR | Zbl

[2] W.E. Aubry, Mather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math. 52 (1999) 811-828. | MR | Zbl

[3] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997). | MR | Zbl

[4] U. Bessi, An analytic counterexample to the KAM theorem. Ergod. Theory Dyn. Syst. 20 (2000) 317-333. | MR | Zbl

[5] A. Biryuk and D. Gomes, An introduction to the Aubry-Mather theory. São Paulo Journal of Mathematical Sciences (to appear).

[6] G. Contreras, R. Iturriaga, G.P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values. Geom. Funct. Anal. 8 (1998) 788-809. | MR | Zbl

[7] L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI, USA (1998). | MR | Zbl

[8] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157 (2001) 1-33. | MR | Zbl

[9] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch. Ration. Mech. Anal. 161 (2002) 271-305. | MR | Zbl

[10] A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. | MR | Zbl

[11] A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043-1046. | MR | Zbl

[12] A. Fathi, Orbite hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1213-1216. | MR | Zbl

[13] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 267-270. | MR | Zbl

[14] A. Fathi and A. Siconolfi, Existence of $C^{1}$ critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004) 363-388. | MR | Zbl

[15] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (1993). | MR | Zbl

[16] G. Forni, Analytic destruction of invariant circles. Ergod. Theory Dyn. Syst. 14 (1994) 267-298. | MR | Zbl

[17] G. Forni, Construction of invariant measures supported within the gaps of Aubry-Mather sets. Ergod. Theory Dyn. Syst. 16 (1996) 51-86. | MR | Zbl

[18] H. Goldstein, Classical mechanics. Addison-Wesley Publishing Co., Reading, Mass., second edition (1980). | MR | Zbl

[19] D.A. Gomes, Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems. Calc. Var. Partial Differential Equations 14 (2002) 345-357. | MR | Zbl

[20] D.A. Gomes, Perturbation theory for viscosity solutions of Hamilton-Jacobi equations and stability of Aubry-Mather sets. SIAM J. Math. Anal. 35 (2003) 135-147 (electronic). | MR | Zbl

[21] D.A. Gomes, Duality principles for fully nonlinear elliptic equations, in Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser, Basel (2005) 125-136. | MR

[22] D.A. Gomes and A.M. Oberman, Computing the effective Hamiltonian using a variational approach. SIAM J. Contr. Opt. 43 (2004) 792-812 (electronic). | MR | Zbl

[23] D.A. Gomes and E. Valdinoci, Lack of integrability via viscosity solution methods. Indiana Univ. Math. J. 53 (2004) 1055-1071. | MR | Zbl

[24] À. Haro, Converse KAM theory for monotone positive symplectomorphisms. Nonlinearity 12 (1999) 1299-1322. | MR | Zbl

[25] A. Knauf, Closed orbits and converse KAM theory. Nonlinearity 3 (1990) 961-973. | MR | Zbl

[26] P.L. Lions and P. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Math. Appl. 56 (2003) 1501-1524. | MR | Zbl

[27] P.L. Lions, G. Papanicolao and S.R.S. Varadhan, Homogeneization of Hamilton-Jacobi equations. Preliminary version (1988).

[28] R.S. Mackay, Converse KAM theory, in Singular behavior and nonlinear dynamics, Vol. 1 (Sámos, 1988), World Sci. Publishing, Teaneck, USA (1989) 109-113. | MR

[29] R.S. Mackay and I.C. Percival, Converse KAM: theory and practice. Comm. Math. Phys. 98 (1985) 469-512. | MR | Zbl

[30] R.S. Mackay, J.D. Meiss and J. Stark, Converse KAM theory for symplectic twist maps. Nonlinearity 2 (1989) 555-570. | MR | Zbl

[31] R. Mañé, On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5 (1992) 623-638. | MR | Zbl

[32] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996) 273-310. | MR | Zbl

[33] J.N. Mather, Minimal action measures for positive-definite Lagrangian systems, in IXth International Congress on Mathematical Physics (Swansea, 1988), Hilger, Bristol (1989) 466-468. | MR | Zbl

[34] J.N. Mather, Minimal measures. Comment. Math. Helv. 64 (1989) 375-394. | MR | Zbl

[35] J.N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. | MR | Zbl

[36] J. Qian, Two approximations for effective hamiltonians arising from homogenization of Hamilton-Jacobi equations. Preprint (2003). | MR

Cité par Sources :