Advanced Search
Article Contents
Article Contents

Gevrey normal form and effective stability of Lagrangian tori

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary 70H08; Secondary: 53C35.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps, Annales de l'Institut Fourier, 45 (1995), 859-895.


    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.


    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.


    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.


    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.


    V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' Springer-Verlag, Berlin, 1993.


    J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20, Dunod, Paris, 1970.


    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.


    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.


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


    F. W. J. Olver, "Asymptotics and Special Functions,'' Academic Press, New York - London, 1974.


    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.


    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.


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


    G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.


    G. Popov and P. TopalovInvariants of isospectral deformations and spectral rigidity, preprint, arXiv:0906.0449v1.


    F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dyn. Syst., 18 (2003), 159-163.doi: 10.1080/1468936031000117857.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(115) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint