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December  2010, 3(4): 643-666. doi: 10.3934/dcdss.2010.3.643

## Gevrey normal form and effective stability of Lagrangian tori

 1 University of Rousse, Department of Algebra and Geometry, 7012, Rousse, Bulgaria 2 Université de Nantes, Laboratoire de mathématiques Jean Leray, 2, rue de la Houssinière, BP 92208, 44072 Nantes Cedex 03, France

Received  April 2009 Revised  June 2010 Published  August 2010

A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.
Citation: Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
 [1] Sh. Alimov, R. Ashurov and A. Pulatov, Multiple Fourier series and Fourier integrals, commutative harmonic analysis IV, Encyclopaedia Math. Sci., 42, 1-95, Springer, Berlin, 1992. [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. [3] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [4] A. Giorgilli and A. Morbidelli, Invariant KAM tori and global stability for Hamiltonian systems, Z. Angew. Math. Phys., 48 (1997), 102-134. doi: 10.1007/PL00001462. [5] T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps, Annales de l'Institut Fourier, 45 (1995), 859-895. [6] G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, 212 (2005), 1-61. doi: 10.1016/j.jde.2004.10.015. [7] G. Iooss and E. Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $0^{2+}$$i\omega resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. [8] M. Herman, Inégalités "a priori'' pour des tores lagrangiens invariants par des difféomor-phismes symplectiques, (French) [A priori inequalities for Lagrangian tori invariant under symplectic diffeomorphisms], Publ. Math. Inst. Hautes Études Sci., 70 (1989), 47-101. [9] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 69-72. doi: 10.3792/pjaa.55.69. [10] V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' Springer-Verlag, Berlin, 1993. [11] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20, Dunod, Paris, 1970. [12] J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199-275. [13] A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems, Phys. D, 86 (1995), 514-516. doi: 10.1016/0167-2789(95)00199-E. [14] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145. [15] F. W. J. Olver, "Asymptotics and Special Functions,'' Academic Press, New York - London, 1974. [16] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 223-248. doi: 10.1007/PL00001004. [17] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms II - Quantum Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 249-279. doi: 10.1007/PL00001005. [18] G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458. [19] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107. [20] G. Popov and P. Topalov, Invariants of isospectral deformations and spectral rigidity, preprint, arXiv:0906.0449v1. [21] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dyn. Syst., 18 (2003), 159-163. doi: 10.1080/1468936031000117857. [22] J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition, J. Differential Equations, 235 (2007), 609-622. doi: 10.1016/j.jde.2006.12.001. show all references ##### References:  [1] Sh. Alimov, R. Ashurov and A. Pulatov, Multiple Fourier series and Fourier integrals, commutative harmonic analysis IV, Encyclopaedia Math. Sci., 42, 1-95, Springer, Berlin, 1992. [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. [3] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [4] A. Giorgilli and A. Morbidelli, Invariant KAM tori and global stability for Hamiltonian systems, Z. Angew. Math. Phys., 48 (1997), 102-134. doi: 10.1007/PL00001462. [5] T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps, Annales de l'Institut Fourier, 45 (1995), 859-895. [6] G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, 212 (2005), 1-61. doi: 10.1016/j.jde.2004.10.015. [7] G. Iooss and E. Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible 0^{2+}$$i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. [8] M. Herman, Inégalités "a priori'' pour des tores lagrangiens invariants par des difféomor-phismes symplectiques, (French) [A priori inequalities for Lagrangian tori invariant under symplectic diffeomorphisms], Publ. Math. Inst. Hautes Études Sci., 70 (1989), 47-101. [9] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 69-72. doi: 10.3792/pjaa.55.69. [10] V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' Springer-Verlag, Berlin, 1993. [11] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20, Dunod, Paris, 1970. [12] J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199-275. [13] A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems, Phys. D, 86 (1995), 514-516. doi: 10.1016/0167-2789(95)00199-E. [14] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145. [15] F. W. J. Olver, "Asymptotics and Special Functions,'' Academic Press, New York - London, 1974. [16] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 223-248. doi: 10.1007/PL00001004. [17] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms II - Quantum Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 249-279. doi: 10.1007/PL00001005. [18] G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458. [19] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107. [20] G. Popov and P. Topalov, Invariants of isospectral deformations and spectral rigidity, preprint, arXiv:0906.0449v1. [21] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dyn. Syst., 18 (2003), 159-163. doi: 10.1080/1468936031000117857. [22] J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition, J. Differential Equations, 235 (2007), 609-622. doi: 10.1016/j.jde.2006.12.001.
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