Gevrey normal form and effective stability of Lagrangian tori

Previous Article
Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
DCDS-S Home
This Issue
Next Article
Finite smooth normal forms and integrability of local families of vector fields

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

Abstract
Related Papers

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 (1992), 1.
[2]	A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (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. 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. 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.
[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.
[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.
[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.
[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.
[10]	V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (1993).
[11]	J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20,, Dunod, (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.
[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.
[14]	A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori,, J. Statist. Phys., 78 (1995), 1607. doi: 10.1007/BF02180145.
[15]	F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (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. 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. doi: 10.1007/PL00001005.
[18]	G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753. doi: 10.1017/S0143385704000458.
[19]	G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.
[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. 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. 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 (1992), 1.
[2]	A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (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. 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. 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.
[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.
[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.
[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.
[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.
[10]	V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (1993).
[11]	J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' (French), Vol. 3, Travaux et recherches mathématiques 20,, Dunod, (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.
[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.
[14]	A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori,, J. Statist. Phys., 78 (1995), 1607. doi: 10.1007/BF02180145.
[15]	F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (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. 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. doi: 10.1007/PL00001005.
[18]	G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753. doi: 10.1017/S0143385704000458.
[19]	G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.
[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. 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. doi: 10.1016/j.jde.2006.12.001.

[1]	Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
[2]	Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[3]	Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[4]	Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941
[5]	Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529
[6]	Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[7]	Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[8]	Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[9]	P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677
[10]	Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[11]	Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[12]	John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[13]	Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[14]	Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[15]	Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612
[16]	Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701
[17]	Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[18]	Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[19]	Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357
[20]	Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences