Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T07:13:57.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscosity solutions methods for converse KAM theory

Published online by Cambridge University Press:  25 September 2008

Diogo A. Gomes  and
Adam Oberman
Diogo A. Gomes
Affiliation:
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. dgomes@math.ist.utl.pt
Adam Oberman
Affiliation:
Instituto Superior Tecnico, Department of Mathematics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. dgomes@math.ist.utl.pt
Get access

Abstract

The main objective of this paper is to provenew necessary conditions to the existence ofKAM tori. To do so, we develop a set ofexplicit a-priori estimates for smoothsolutions of Hamilton-Jacobi equations,using a combination of methods fromviscosity solutions,KAM and Aubry-Mather theories.These estimatesare validin anyspace dimension, and can be checked numericallyto 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, andthe double pendulum.

Keywords

Aubry-Mather theoryHamilton-Jacobi integrabilityviscosity solutions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 1047 - 1064
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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].
Aubry, W.E., Mather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math. 52 (1999) 811–828.
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997).
Bessi, U., An analytic counterexample to the KAM theorem. Ergod. Theory Dyn. Syst. 20 (2000) 317–333. CrossRef
A. Biryuk and D. Gomes, An introduction to the Aubry-Mather theory. São Paulo Journal of Mathematical Sciences (to appear).
Contreras, G., Iturriaga, R., Paternain, G.P. and Paternain, M., Lagrangian graphs, minimizing measures and Mañé's critical values. Geom. Funct. Anal. 8 (1998) 788–809. CrossRef
L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI, USA (1998).
Evans, L.C. and Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157 (2001) 1–33. CrossRef
Evans, L.C. and Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch. Ration. Mech. Anal. 161 (2002) 271–305. CrossRef
Fathi, A., Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649–652. CrossRef
Fathi, A., Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043–1046. CrossRef
Fathi, A., Orbite hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1213–1216. CrossRef
Fathi, A., Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 267–270. CrossRef
Fathi, A. and Siconolfi, A., Existence of $C\sp 1$ critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004) 363–388. CrossRef
W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (1993).
Forni, G., Analytic destruction of invariant circles. Ergod. Theory Dyn. Syst. 14 (1994) 267–298. CrossRef
Forni, G., Construction of invariant measures supported within the gaps of Aubry-Mather sets. Ergod. Theory Dyn. Syst. 16 (1996) 51–86. CrossRef
H. Goldstein, Classical mechanics. Addison-Wesley Publishing Co., Reading, Mass., second edition (1980).
Gomes, D.A., Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems. Calc. Var. Partial Differential Equations 14 (2002) 345–357. CrossRef
Gomes, D.A., Perturbation theory for viscosity solutions of Hamilton-Jacobi equations and stability of Aubry-Mather sets. SIAM J. Math. Anal. 35 (2003) 135–147 (electronic). CrossRef
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.
Gomes, D.A. and Oberman, A.M., Computing the effective Hamiltonian using a variational approach. SIAM J. Contr. Opt. 43 (2004) 792–812 (electronic). CrossRef
Gomes, D.A. and Valdinoci, E., Lack of integrability via viscosity solution methods. Indiana Univ. Math. J. 53 (2004) 1055–1071. CrossRef
Haro, À., Converse KAM theory for monotone positive symplectomorphisms. Nonlinearity 12 (1999) 1299–1322. CrossRef
Knauf, A., Closed orbits and converse KAM theory. Nonlinearity 3 (1990) 961–973. CrossRef
Lions, P.L. and Souganidis, P., Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Math. Appl. 56 (2003) 1501–1524. CrossRef
P.L. Lions, G. Papanicolao and S.R.S. Varadhan, Homogeneization of Hamilton-Jacobi equations. Preliminary version (1988).
R.S. MacKay, Converse KAM theory, in Singular behavior and nonlinear dynamics, Vol. 1 (Sámos, 1988), World Sci. Publishing, Teaneck, USA (1989) 109–113.
MacKay, R.S. and Percival, I.C., Converse KAM: theory and practice. Comm. Math. Phys. 98 (1985) 469–512. CrossRef
MacKay, R.S., Meiss, J.D. and Stark, J., Converse KAM theory for symplectic twist maps. Nonlinearity 2 (1989) 555–570. CrossRef
Mañé, R., On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5 (1992) 623–638. CrossRef
Mañé, R., Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996) 273–310. CrossRef
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.
Mather, J.N., Minimal measures. Comment. Math. Helv. 64 (1989) 375–394. CrossRef
Mather, J.N., Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169–207. CrossRef
J. Qian, Two approximations for effective hamiltonians arising from homogenization of Hamilton-Jacobi equations. Preprint (2003).