American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Finite smooth normal forms and integrability of local families of vector fields
  • DCDS-S Home
  • This Issue
  • Next Article
    Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
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 (1992), 1.   Google Scholar

[2]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (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.  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.  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.   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.  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.   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.   Google Scholar

[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.  Google Scholar

[10]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (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, (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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (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.  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.  doi: 10.1007/PL00001005.  Google Scholar

[18]

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

[19]

G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.   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.  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.  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 (1992), 1.   Google Scholar

[2]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, "Higher Transcendental Functions,'' Vols. I, II,, McGraw-Hill Book Company, (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.  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.  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.   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.  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.   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.   Google Scholar

[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.  Google Scholar

[10]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'', Springer-Verlag, (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, (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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

F. W. J. Olver, "Asymptotics and Special Functions,'', Academic Press, (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.  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.  doi: 10.1007/PL00001005.  Google Scholar

[18]

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

[19]

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

[1]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[2]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[3]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[4]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
[6]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[7]

John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004
[8]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[9]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[10]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[11]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[12]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[13]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[14]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[15]

Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069
[16]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066
[17]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068
[18]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
[19]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences