Gevrey normal form and effective stability of Lagrangian tori

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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

Abstract
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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 and 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.
[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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