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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

Todor Mitev 1, and  Georgi Popov 2,

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.
Keywords: Gevrey classes., effective stability, Kronecker tori, KAM theory, Birkhoff normal form.
Mathematics Subject Classification: Primary 70H08; Secondary: 53C3.
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
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.  Google Scholar

[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.  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), 167-198. 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), 102-134. 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), 859-895.  Google Scholar

[6]

G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, 212 (2005), 1-61. 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), 831-838.  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), 47-101.  Google Scholar

[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.  Google Scholar

[10]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' Springer-Verlag, Berlin, 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, Paris, 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), 199-275.  Google Scholar

[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.  Google Scholar

[14]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145.  Google Scholar

[15]

F. W. J. Olver, "Asymptotics and Special Functions,'' Academic Press, New York - London, 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), 223-248. 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), 249-279. doi: 10.1007/PL00001005.  Google Scholar

[18]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.  Google Scholar

[19]

G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.  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), 159-163. 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), 609-622. 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, 1-95, Springer, Berlin, 1992.  Google Scholar

[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.  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), 167-198. 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), 102-134. 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), 859-895.  Google Scholar

[6]

G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, 212 (2005), 1-61. 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), 831-838.  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), 47-101.  Google Scholar

[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.  Google Scholar

[10]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' Springer-Verlag, Berlin, 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, Paris, 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), 199-275.  Google Scholar

[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.  Google Scholar

[14]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145.  Google Scholar

[15]

F. W. J. Olver, "Asymptotics and Special Functions,'' Academic Press, New York - London, 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), 223-248. 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), 249-279. doi: 10.1007/PL00001005.  Google Scholar

[18]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.  Google Scholar

[19]

G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.  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), 159-163. 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), 609-622. doi: 10.1016/j.jde.2006.12.001.  Google Scholar

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