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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 |
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. |
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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