Generic Nekhoroshev theory without small divisors
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, p. 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 : https://doi.org/10.5802/aif.2706
Classification:  37J25,  37J40,  70H08,  70H09,  70K45,  70K60,  70K65
Keywords: Hamiltonian systems, perturbation of integrable systems, effective stability
@article{AIF_2012__62_1_277_0,
     author = {Bounemoura, Abed and Niederman, Laurent},
     title = {Generic Nekhoroshev theory without small divisors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     pages = {277-324},
     doi = {10.5802/aif.2706},
     zbl = {1257.37036},
     mrnumber = {2986272},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_1_277_0}
}
Generic Nekhoroshev theory without small divisors. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 277-324. doi : 10.5802/aif.2706. https://aif.centre-mersenne.org/item/AIF_2012__62_1_277_0/

[1] Arnold, Vladimir I. Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, Tome 156 (1964), pp. 9-12 | MR 163026 | Zbl 0135.42602

[2] Arnold, Vladimir I.; Kozlov, Valery V.; Neishtadt, Anatoly I. Mathematical aspects of classical and celestial mechanics, Springer-Verlag, Berlin, Encyclopaedia of Mathematical Sciences, Tome 3 (2006) ([Dynamical systems. III], Translated from the Russian original by E. Khukhro) | MR 2269239 | Zbl 0885.70001

[3] Bambusi, Dario Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., Tome 230 (1999) no. 2, pp. 345-387 | Article | MR 1676714 | Zbl 0928.35160

[4] Bambusi, Dario; Giorgilli, Antonio Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems, J. Statist. Phys., Tome 71 (1993) no. 3-4, pp. 569-606 | Article | MR 1219023 | Zbl 0943.82549

[5] Bambusi, Dario; Nekhoroshev, N. N. Long time stability in perturbations of completely resonant PDE’s, Acta Appl. Math., Tome 70 (2002) no. 1-3, pp. 1-22 (Symmetry and perturbation theory) | Article | MR 1892373

[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 0602.58021

[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, Tome 249 (2010) no. 11, pp. 2905-2920 | Article | MR 2718671

[9] Bourgain, Jean Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems, Tome 24 (2004) no. 5, pp. 1331-1357 | Article | MR 2104588

[10] Cassels, J. W. S. An introduction to Diophantine approximation, Cambridge University Press, New York, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45 (1957) | MR 87708 | Zbl 0077.04801

[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), Tome 13 (1972), p. 255-260 (1973) | MR 326293 | Zbl 0249.43002

[12] Delshams, Amadeu; Gutiérrez, Pere Effective stability and KAM theory, J. Differential Equations, Tome 128 (1996) no. 2, pp. 415-490 | Article | MR 1398328 | Zbl 0858.58012

[13] Fassò, Francesco; Guzzo, Massimiliano; Benettin, Giancarlo Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Comm. Math. Phys., Tome 197 (1998) no. 2, pp. 347-360 | Article | MR 1652750 | Zbl 0928.37017

[14] Hunt, Brian R.; Kaloshin, Vadim Yu.; Henk Broer, F Takens; Hasselblatt, B Prevalence, Elsevier Science (Handbook of Dynamical Systems) Tome 3 (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.), Tome 27 (1992) no. 2, pp. 217-238 | Article | MR 1161274 | Zbl 0763.28009

[16] Il’Yashenko, Yu. S. A criterion of steepness for analytic functions, Uspekhi Mat. Nauk, Tome 41 (1986) no. 1(247), p. 193-194 | MR 832421 | Zbl 0597.32003

[17] Khanin, Kostya; Lopes Dias, João; Marklof, Jens Renormalization of multidimensional Hamiltonian flows, Nonlinearity, Tome 19 (2006) no. 12, pp. 2727-2753 | Article | MR 2273756

[18] Khanin, Kostya; Lopes Dias, João; Marklof, Jens Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., Tome 270 (2007) no. 1, pp. 197-231 | Article | MR 2276445

[19] Kolmogorov, A. N. On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.), Tome 98 (1954), pp. 527-530 | MR 68687 | Zbl 0056.31502

[20] De La Llave, Rafael A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 69 (2001), pp. 175-292 | MR 1858536

[21] Lochak, P. Canonical perturbation theory: an approach based on joint approximations, Uspekhi Mat. Nauk, Tome 47 (1992) no. 6(288), pp. 59-140 | MR 1209145 | Zbl 0795.58042

[22] Lochak, P.; Meunier, C. Multiphase averaging for classical systems, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 72 (1988) (With applications to adiabatic theorems, Translated from the French by H. S. Dumas) | MR 959890 | Zbl 0668.34044

[23] Lochak, P.; Neĭshtadt, A. I. Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, Tome 2 (1992) no. 4, pp. 495-499 | Article | MR 1195881 | Zbl 1055.37573

[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), Birkhäuser, Basel (Progr. Nonlinear Differential Equations Appl.) Tome 12 (1994), pp. 15-34 | MR 1279386 | Zbl 0807.70020

[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 1986314

[26] Morbidelli, Alessandro; Giorgilli, Antonio Superexponential stability of KAM tori, J. Statist. Phys., Tome 78 (1995) no. 5-6, pp. 1607-1617 | Article | MR 1316113 | Zbl 1080.37512

[27] Morbidelli, Alessandro; Guzzo, Massimiliano The Nekhoroshev theorem and the asteroid belt dynamical system, Celestial Mech. Dynam. Astronom., Tome 65 (1996/97) no. 1-2, pp. 107-136 (The dynamical behaviour of our planetary system (Ramsau, 1996)) | Article | MR 1461601 | Zbl 0891.70007

[28] Neĭshtadt, A. I. The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., Tome 48 (1984) no. 2, pp. 197-204 | MR 802878 | Zbl 0571.70022

[29] Nekhorošev, N. N. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, Tome 32 (1977) no. 6(198), p. 5-66, 287 | MR 501140 | Zbl 0389.70028

[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 549621 | Zbl 0668.34046

[31] Niederman, Laurent Nonlinear stability around an elliptic equilibrium point in a Hamiltonian system, Nonlinearity, Tome 11 (1998) no. 6, pp. 1465-1479 | Article | MR 1660357 | Zbl 0917.58015

[32] Niederman, Laurent Exponential stability for small perturbations of steep integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, Tome 24 (2004) no. 2, pp. 593-608 | Article | MR 2054052 | Zbl 1071.37038

[33] Niederman, Laurent Hamiltonian stability and subanalytic geometry, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 3, pp. 795-813 | Article | Numdam | MR 2244230 | Zbl 1120.14048

[34] Niederman, Laurent Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, Tome 27 (2007) no. 3, pp. 905-928 | Article | MR 2322185 | Zbl 1130.37386

[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.), Tome 42 (2005) no. 3, p. 263-290 (electronic) | Article | MR 2149086 | Zbl 1111.28014

[37] Pöschel, Jürgen On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, Tome 12 (1999) no. 6, pp. 1587-1600 | Article | MR 1726666 | Zbl 0935.35154

[38] Pöschel, Jürgen On Nekhoroshev’s estimate at an elliptic equilibrium, Internat. Math. Res. Notices (1999) no. 4, pp. 203-215 | Article | Zbl 0918.58026

[39] Pöschel, Jürgen A lecture on the classical KAM theorem, Smooth ergodic theory and its applications (Seattle, WA, 1999), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 69 (2001), pp. 707-732 | MR 1858551

[40] Ramis, Jean-Pierre; Schäfke, Reinhard Gevrey separation of fast and slow variables, Nonlinearity, Tome 9 (1996) no. 2, pp. 353-384 | Article | MR 1384480 | Zbl 0925.70161

[41] Yomdin, Y. The geometry of critical and near-critical values of differentiable mappings, Math. Ann., Tome 264 (1983) no. 4, pp. 495-515 | Article | MR 716263 | Zbl 0507.57019

[42] Yomdin, Yosef; Comte, Georges Tame geometry with application in smooth analysis, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1834 (2004) | MR 2041428