Generic Nekhoroshev theory without small divisors
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 277-324.

In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

Dans cet article, nous présentons une nouvelle approche de la théorie de Nekhoroshev pour un hamiltonien intégrable générique, qui évite complètement le problème des petits diviseurs. La preuve est une extension d’une méthode introduite par Lochak, elle n’utilise que des moyennisations périodiques et de l’approximation diophantienne simultanée, ainsi que des arguments géométriques introduit par le second auteur. Notre méthode permet également d’obtenir des résultats de stabilité pour des hamiltoniens génériques non-analytiques, ainsi que de nouveaux résultats de stabilité au voisinage des tores invariants linéairement stables.

DOI: 10.5802/aif.2706
Classification: 37J25, 37J40, 70H08, 70H09, 70K45, 70K60, 70K65
Keywords: Hamiltonian systems, perturbation of integrable systems, effective stability
Mot clés : systèmes hamiltoniens, perturbation de systèmes intégrables, stabilité effective
Bounemoura, Abed 1; Niederman, Laurent 1

1 Université Paris-Sud 11 Faculté des Sciences d’Orsay Département de Mathématiques Bâtiment 425 91405 Orsay cedex (France)
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Bounemoura, Abed; Niederman, Laurent. Generic Nekhoroshev theory without small divisors. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 277-324. doi : 10.5802/aif.2706. http://www.numdam.org/articles/10.5802/aif.2706/

[1] Arnold, Vladimir I. Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, Volume 156 (1964), pp. 9-12 | MR | Zbl

[2] Arnold, Vladimir I.; Kozlov, Valery V.; Neishtadt, Anatoly I. Mathematical aspects of classical and celestial mechanics, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006 ([Dynamical systems. III], Translated from the Russian original by E. Khukhro) | MR | Zbl

[3] Bambusi, Dario Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., Volume 230 (1999) no. 2, pp. 345-387 | DOI | MR | Zbl

[4] Bambusi, Dario; Giorgilli, Antonio Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems, J. Statist. Phys., Volume 71 (1993) no. 3-4, pp. 569-606 | DOI | MR | Zbl

[5] Bambusi, Dario; Nekhoroshev, N. N. Long time stability in perturbations of completely resonant PDE’s, Acta Appl. Math., Volume 70 (2002) no. 1-3, pp. 1-22 (Symmetry and perturbation theory) | DOI | MR

[6] Bost, Jean-Benoît Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolmogorov, Arnol d, Moser, Rüssmann, Zehnder, Herman, Pöschel,...), Astérisque (1986) no. 133-134, pp. 113-157 (Seminar Bourbaki, Vol. 1984/85) | Numdam | Zbl

[7] Bounemoura, Abed Generic super-exponential stability of invariant tori (2009) (to appear)

[8] Bounemoura, Abed Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, J. Differential Equations, Volume 249 (2010) no. 11, pp. 2905-2920 | DOI | MR

[9] Bourgain, Jean Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 5, pp. 1331-1357 | DOI | MR

[10] Cassels, J. W. S. An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957 | MR | Zbl

[11] Christensen, Jens Peter Reus On sets of Haar measure zero in abelian Polish groups, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Volume 13 (1972), p. 255-260 (1973) | MR | Zbl

[12] Delshams, Amadeu; Gutiérrez, Pere Effective stability and KAM theory, J. Differential Equations, Volume 128 (1996) no. 2, pp. 415-490 | DOI | MR | Zbl

[13] Fassò, Francesco; Guzzo, Massimiliano; Benettin, Giancarlo Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Comm. Math. Phys., Volume 197 (1998) no. 2, pp. 347-360 | DOI | MR | Zbl

[14] Hunt, Brian R.; Kaloshin, Vadim Yu.; Henk Broer, F Takens; Hasselblatt, B Prevalence (Handbook of Dynamical Systems), Volume 3, Elsevier Science, 2010, pp. 43 -87

[15] Hunt, Brian R.; Sauer, Tim; Yorke, James A. Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 2, pp. 217-238 | DOI | MR | Zbl

[16] Il’yashenko, Yu. S. A criterion of steepness for analytic functions, Uspekhi Mat. Nauk, Volume 41 (1986) no. 1(247), pp. 193-194 | MR | Zbl

[17] Khanin, Kostya; Lopes Dias, João; Marklof, Jens Renormalization of multidimensional Hamiltonian flows, Nonlinearity, Volume 19 (2006) no. 12, pp. 2727-2753 | DOI | MR

[18] Khanin, Kostya; Lopes Dias, João; Marklof, Jens Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., Volume 270 (2007) no. 1, pp. 197-231 | DOI | MR

[19] Kolmogorov, A. N. On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.), Volume 98 (1954), pp. 527-530 | MR | Zbl

[20] de la Llave, Rafael A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proc. Sympos. Pure Math.), Volume 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175-292 | MR

[21] Lochak, P. Canonical perturbation theory: an approach based on joint approximations, Uspekhi Mat. Nauk, Volume 47 (1992) no. 6(288), pp. 59-140 | MR | Zbl

[22] Lochak, P.; Meunier, C. Multiphase averaging for classical systems, Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988 (With applications to adiabatic theorems, Translated from the French by H. S. Dumas) | MR | Zbl

[23] Lochak, P.; Neĭshtadt, A. I. Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, Volume 2 (1992) no. 4, pp. 495-499 | DOI | MR | Zbl

[24] Lochak, P.; Neĭshtadt, A. I.; Niederman, L. Stability of nearly integrable convex Hamiltonian systems over exponentially long times, Seminar on Dynamical Systems (St. Petersburg, 1991) (Progr. Nonlinear Differential Equations Appl.), Volume 12, Birkhäuser, Basel, 1994, pp. 15-34 | MR | Zbl

[25] Marco, Jean-Pierre; Sauzin, David Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci. (2002) no. 96, p. 199-275 (2003) | Numdam | MR

[26] Morbidelli, Alessandro; Giorgilli, Antonio Superexponential stability of KAM tori, J. Statist. Phys., Volume 78 (1995) no. 5-6, pp. 1607-1617 | DOI | MR | Zbl

[27] Morbidelli, Alessandro; Guzzo, Massimiliano The Nekhoroshev theorem and the asteroid belt dynamical system, Celestial Mech. Dynam. Astronom., Volume 65 (1996/97) no. 1-2, pp. 107-136 The dynamical behaviour of our planetary system (Ramsau, 1996) | DOI | MR | Zbl

[28] Neĭshtadt, A. I. The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., Volume 48 (1984) no. 2, pp. 197-204 | MR | Zbl

[29] Nekhorošev, N. N. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, Volume 32 (1977) no. 6(198), p. 5-66, 287 | MR | Zbl

[30] Nekhorošev, N. N. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk. (1979) no. 5, pp. 5-50 | MR | Zbl

[31] Niederman, Laurent Nonlinear stability around an elliptic equilibrium point in a Hamiltonian system, Nonlinearity, Volume 11 (1998) no. 6, pp. 1465-1479 | DOI | MR | Zbl

[32] Niederman, Laurent Exponential stability for small perturbations of steep integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 2, pp. 593-608 | DOI | MR | Zbl

[33] Niederman, Laurent Hamiltonian stability and subanalytic geometry, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 3, pp. 795-813 | DOI | EuDML | Numdam | MR | Zbl

[34] Niederman, Laurent Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, Volume 27 (2007) no. 3, pp. 905-928 | DOI | MR | Zbl

[35] Niederman, Laurent Nekhoroshev Theory, Encyclopedia of Complexity and Systems Science, Springer, 2009, pp. 5986-5998

[36] Ott, William; Yorke, James A. Prevalence, Bull. Amer. Math. Soc. (N.S.), Volume 42 (2005) no. 3, p. 263-290 (electronic) | DOI | MR | Zbl

[37] Pöschel, Jürgen On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, Volume 12 (1999) no. 6, pp. 1587-1600 | DOI | MR | Zbl

[38] Pöschel, Jürgen On Nekhoroshev’s estimate at an elliptic equilibrium, Internat. Math. Res. Notices (1999) no. 4, pp. 203-215 | DOI | Zbl

[39] Pöschel, Jürgen A lecture on the classical KAM theorem, Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proc. Sympos. Pure Math.), Volume 69, Amer. Math. Soc., Providence, RI, 2001, pp. 707-732 | MR

[40] Ramis, Jean-Pierre; Schäfke, Reinhard Gevrey separation of fast and slow variables, Nonlinearity, Volume 9 (1996) no. 2, pp. 353-384 | DOI | MR | Zbl

[41] Yomdin, Y. The geometry of critical and near-critical values of differentiable mappings, Math. Ann., Volume 264 (1983) no. 4, pp. 495-515 | DOI | MR | Zbl

[42] Yomdin, Yosef; Comte, Georges Tame geometry with application in smooth analysis, Lecture Notes in Mathematics, 1834, Springer-Verlag, Berlin, 2004 | MR

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