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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 & 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 (1992), 1.   Google Scholar [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (1953).   Google Scholar [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.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar [4] A. Giorgilli and A. Morbidelli, Invariant KAM tori and global stability for Hamiltonian systems, , Z. Angew. Math. Phys., 48 (1997), 102.  doi: 10.1007/PL00001462.  Google Scholar [5] T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps,, Annales de l'Institut Fourier, 45 (1995), 859.   Google Scholar [6] G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields,, J. Differential Equations, 212 (2005), 1.  doi: 10.1016/j.jde.2004.10.015.  Google Scholar [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. Google Scholar [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. Google Scholar [9] H. Komatsu, The implicit function theorem for ultradifferentiable mappings,, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 69. doi: 10.3792/pjaa.55.69. Google Scholar [10] V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (1993). Google Scholar [11] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20,, Dunod, (1970). Google Scholar [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. Google Scholar [13] A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems,, Phys. D, 86 (1995), 514. doi: 10.1016/0167-2789(95)00199-E. Google Scholar [14] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori,, J. Statist. Phys., 78 (1995), 1607. doi: 10.1007/BF02180145. Google Scholar [15] F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (1974). Google Scholar [16] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223. doi: 10.1007/PL00001004. Google Scholar [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. doi: 10.1007/PL00001005. Google Scholar [18] G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753. doi: 10.1017/S0143385704000458. Google Scholar [19] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87. Google Scholar [20] G. Popov and P. Topalov, Invariants of isospectral deformations and spectral rigidity, preprint,, \arXiv{0906.0449v1}., (). Google Scholar [21] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159. doi: 10.1080/1468936031000117857. Google Scholar [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. doi: 10.1016/j.jde.2006.12.001. Google Scholar 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 (1992), 1. Google Scholar [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (1953). Google Scholar [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. doi: 10.1016/0022-0396(89)90161-7. Google Scholar [4] A. Giorgilli and A. Morbidelli, Invariant KAM tori and global stability for Hamiltonian systems, , Z. Angew. Math. Phys., 48 (1997), 102. doi: 10.1007/PL00001462. Google Scholar [5] T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps,, Annales de l'Institut Fourier, 45 (1995), 859. Google Scholar [6] G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields,, J. Differential Equations, 212 (2005), 1. doi: 10.1016/j.jde.2004.10.015. Google Scholar [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.   Google Scholar [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.   Google Scholar [9] H. Komatsu, The implicit function theorem for ultradifferentiable mappings,, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 69.  doi: 10.3792/pjaa.55.69.  Google Scholar [10] V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (1993).   Google Scholar [11] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20,, Dunod, (1970).   Google Scholar [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.   Google Scholar [13] A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems,, Phys. D, 86 (1995), 514.  doi: 10.1016/0167-2789(95)00199-E.  Google Scholar [14] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori,, J. Statist. Phys., 78 (1995), 1607.  doi: 10.1007/BF02180145.  Google Scholar [15] F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (1974).   Google Scholar [16] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223.  doi: 10.1007/PL00001004.  Google Scholar [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.  doi: 10.1007/PL00001005.  Google Scholar [18] G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753.  doi: 10.1017/S0143385704000458.  Google Scholar [19] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.   Google Scholar [20] G. Popov and P. Topalov, Invariants of isospectral deformations and spectral rigidity, preprint,, \arXiv{0906.0449v1}., ().   Google Scholar [21] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159.  doi: 10.1080/1468936031000117857.  Google Scholar [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.  doi: 10.1016/j.jde.2006.12.001.  Google Scholar
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