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KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character
A parametrised version of Moser's modifying terms theorem
1. | CeNDEF, Dept. of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, Netherlands |
References:
[1] |
V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Springer, (1988).
|
[2] |
V. I. Arnol'd, "Mathematical Methods Of Classical Mechanics,", Springer, (1989).
|
[3] |
R. I. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane,, Trudy Seminara Imeni I.G. Petrovskogo, 2 (1976), 23.
|
[4] |
R. I. Bogdanov, The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues,, Trudy Seminara Imeni I. G. Petrovskogo, 2 (1976), 37.
|
[5] |
J. Bonet, R. W. Braun, R. Meise and B. A. Taylor, Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions,, Studia Mathematica, 99 (1991), 155.
|
[6] |
H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori,, Physica D, 238 (2009), 309.
doi: 10.1016/j.physd.2008.10.004. |
[7] |
H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation,, International Journal of Bifurcation and Chaos, 17 (2007), 2605.
doi: 10.1142/S021812740701866X. |
[8] |
H. W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems,, Nonlinearity, 18 (2005), 1735.
doi: 10.1088/0951-7715/18/4/018. |
[9] |
H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori,, Journal of Differential Equations, 232 (2007), 355.
doi: 10.1016/j.jde.2006.08.022. |
[10] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems: Order Amidst Chaos,", Lecture Notes in Mathematics, 1645 (1996).
|
[11] |
H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, "Unfoldings and Bifurcations of Quasi-periodic Tori,", Memoirs of the AMS, 83 (1990).
|
[12] |
H. W. Broer, R. Roussarie and C. Simó, On the Bogdanov-Takens bifurcation for planar diffeomorphisms,, Proceedings of Equadiff '91 (C. Perelló, (1993), 81.
|
[13] |
H. W. Broer, R. Roussarie and C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms,, Ergodic Theory and Dynamical Systems, 16 (1996), 1147.
|
[14] |
T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C,0)$,, Bulletin de la Société Mathématique de France, 128 (2000), 69.
|
[15] |
A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes,, Publications Mathématiques de l'I.H.E.S., 61 (1985), 67.
|
[16] |
M. C. Ciocci, "Bifurcation of Periodic Orbits and Persistence of Quasi Periodic Orbits in Families of Reversible Systems,", Ph.D. thesis, (2003). Google Scholar |
[17] |
H. H. de Jong, "Quasi-periodic Breathers in Systems of Weakly Coupled Pendulums: Applications of KAM Theory,", Ph.D. thesis, (1999). Google Scholar |
[18] |
G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension,, Communications of Mathematical Physics, 289 (2009), 863.
doi: 10.1007/s00220-009-0817-1. |
[19] |
A. González-Enríquez, À. Haro and R. de la Llave, Translated torus theory with parameters and applications,, In preparation., (). Google Scholar |
[20] |
M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities,", Graduate Texts in Mathematics, 14 (1973).
|
[21] |
H. Hanßmann, The quasi-periodic centre-saddle bifurcation,, Journal of Differential Equations, 142 (1998), 305.
doi: 10.1006/jdeq.1997.3365. |
[22] |
H. Hanßmann, "Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples,", Lecture Notes in Mathematics, 1893 (2007).
|
[23] |
G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1952).
|
[24] |
J. Hoo, "Quasi-periodic Bifurcations in a Strong Resonance: Combination Tones in Gyroscopic Stabilisation,", Ph.D. thesis, (2005). Google Scholar |
[25] |
L. Hörmander, "An Introduction to Complex Analysis in Several Variables,", North-Holland, (1990).
|
[26] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis,", Springer, (1990).
|
[27] |
X. Li and R. de la Llave, Convergence of differentiable functions on closed sets and remarks on the proofs of the "converse approximation lemmas'',, Preprint 2009., (2009). Google Scholar |
[28] |
J. Moser, Convergent series expansions for quasi-periodic motions,, Mathematische Annalen, 169 (1967), 136.
doi: 10.1007/BF01399536. |
[29] |
V. Poénaru, "Analyse Différentielle,", Lecture Notes in Mathematics, 371 (1974).
|
[30] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I: Birkhoff normal forms,, Annales de l'Institut Henri Poincaré, 1 (2000), 223.
doi: 10.1007/PL00001004. |
[31] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms,, Annales de l'Institut Henri Poincaré, 1 (2000), 249.
|
[32] |
G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753.
doi: 10.1017/S0143385704000458. |
[33] |
J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Communications of Pure and Applied Mathematics, 35 (1982), 653.
doi: 10.1002/cpa.3160350504. |
[34] |
A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space,, Functional Analysis and Applications, 3 (1969), 303.
doi: 10.1007/BF01076316. |
[35] |
R. Roussarie and F. O. O. Wagener, A study of the Bogdanov-Takens bifurcation,, Resenhas IME-USP, 2 (1995), 1.
|
[36] |
H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes,, Nachr. Ak. Wiss. Gött., 5 (1970), 67.
|
[37] |
H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus,, Dynamical Systems, 38 (1975), 598.
|
[38] |
M. B. Sevryuk, KAM tori: Persistence and smoothness,, Nonlinearity, 21 (2008).
doi: 10.1088/0951-7715/21/10/T01. |
[39] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Univ. Press, (1970).
|
[40] |
F. Takens, Forced oscillations and bifurcations,, Communications of the Mathematical Institute 3, (1974), 1.
|
[41] |
F. O. O. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dynamical Systems, 18 (2003), 159.
doi: 10.1080/1468936031000117857. |
[42] |
F. O. O. Wagener, On the quasi-periodic $d$-fold degenerate bifurcation,, Journal of Differential Equations, 216 (2005), 261.
doi: 10.1016/j.jde.2005.06.013. |
[43] |
J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, Journal of Differential Equations, 235 (2007), 609.
doi: 10.1016/j.jde.2006.12.001. |
[44] |
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I,, Communications in Pure and Applied Mathematics, 28 (1975), 91.
|
[45] |
D. Zhang and J. Xu, Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition,, Journal of Mathematical Analysis and Applications, 323 (2006), 293.
doi: 10.1016/j.jmaa.2005.10.029. |
[46] |
D. Zhang and J. Xu, On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Discrete and Continuous Dynamical Systems, 16 (2006), 635.
doi: 10.3934/dcds.2006.16.635. |
[47] |
D. Zhang and J. Xu, Invariant tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Nonlinear Analysis, 67 (2007), 2240.
doi: 10.1016/j.na.2006.09.012. |
[48] |
D. Zhang and J. Xu, Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Acta Mathematica Sinica-English Series, 24 (2008), 1625.
doi: 10.1007/s10114-008-6180-x. |
show all references
References:
[1] |
V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Springer, (1988).
|
[2] |
V. I. Arnol'd, "Mathematical Methods Of Classical Mechanics,", Springer, (1989).
|
[3] |
R. I. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane,, Trudy Seminara Imeni I.G. Petrovskogo, 2 (1976), 23.
|
[4] |
R. I. Bogdanov, The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues,, Trudy Seminara Imeni I. G. Petrovskogo, 2 (1976), 37.
|
[5] |
J. Bonet, R. W. Braun, R. Meise and B. A. Taylor, Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions,, Studia Mathematica, 99 (1991), 155.
|
[6] |
H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori,, Physica D, 238 (2009), 309.
doi: 10.1016/j.physd.2008.10.004. |
[7] |
H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation,, International Journal of Bifurcation and Chaos, 17 (2007), 2605.
doi: 10.1142/S021812740701866X. |
[8] |
H. W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems,, Nonlinearity, 18 (2005), 1735.
doi: 10.1088/0951-7715/18/4/018. |
[9] |
H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori,, Journal of Differential Equations, 232 (2007), 355.
doi: 10.1016/j.jde.2006.08.022. |
[10] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems: Order Amidst Chaos,", Lecture Notes in Mathematics, 1645 (1996).
|
[11] |
H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, "Unfoldings and Bifurcations of Quasi-periodic Tori,", Memoirs of the AMS, 83 (1990).
|
[12] |
H. W. Broer, R. Roussarie and C. Simó, On the Bogdanov-Takens bifurcation for planar diffeomorphisms,, Proceedings of Equadiff '91 (C. Perelló, (1993), 81.
|
[13] |
H. W. Broer, R. Roussarie and C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms,, Ergodic Theory and Dynamical Systems, 16 (1996), 1147.
|
[14] |
T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C,0)$,, Bulletin de la Société Mathématique de France, 128 (2000), 69.
|
[15] |
A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes,, Publications Mathématiques de l'I.H.E.S., 61 (1985), 67.
|
[16] |
M. C. Ciocci, "Bifurcation of Periodic Orbits and Persistence of Quasi Periodic Orbits in Families of Reversible Systems,", Ph.D. thesis, (2003). Google Scholar |
[17] |
H. H. de Jong, "Quasi-periodic Breathers in Systems of Weakly Coupled Pendulums: Applications of KAM Theory,", Ph.D. thesis, (1999). Google Scholar |
[18] |
G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension,, Communications of Mathematical Physics, 289 (2009), 863.
doi: 10.1007/s00220-009-0817-1. |
[19] |
A. González-Enríquez, À. Haro and R. de la Llave, Translated torus theory with parameters and applications,, In preparation., (). Google Scholar |
[20] |
M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities,", Graduate Texts in Mathematics, 14 (1973).
|
[21] |
H. Hanßmann, The quasi-periodic centre-saddle bifurcation,, Journal of Differential Equations, 142 (1998), 305.
doi: 10.1006/jdeq.1997.3365. |
[22] |
H. Hanßmann, "Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples,", Lecture Notes in Mathematics, 1893 (2007).
|
[23] |
G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1952).
|
[24] |
J. Hoo, "Quasi-periodic Bifurcations in a Strong Resonance: Combination Tones in Gyroscopic Stabilisation,", Ph.D. thesis, (2005). Google Scholar |
[25] |
L. Hörmander, "An Introduction to Complex Analysis in Several Variables,", North-Holland, (1990).
|
[26] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis,", Springer, (1990).
|
[27] |
X. Li and R. de la Llave, Convergence of differentiable functions on closed sets and remarks on the proofs of the "converse approximation lemmas'',, Preprint 2009., (2009). Google Scholar |
[28] |
J. Moser, Convergent series expansions for quasi-periodic motions,, Mathematische Annalen, 169 (1967), 136.
doi: 10.1007/BF01399536. |
[29] |
V. Poénaru, "Analyse Différentielle,", Lecture Notes in Mathematics, 371 (1974).
|
[30] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I: Birkhoff normal forms,, Annales de l'Institut Henri Poincaré, 1 (2000), 223.
doi: 10.1007/PL00001004. |
[31] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms,, Annales de l'Institut Henri Poincaré, 1 (2000), 249.
|
[32] |
G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory and Dynamical Systems, 24 (2004), 1753.
doi: 10.1017/S0143385704000458. |
[33] |
J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Communications of Pure and Applied Mathematics, 35 (1982), 653.
doi: 10.1002/cpa.3160350504. |
[34] |
A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space,, Functional Analysis and Applications, 3 (1969), 303.
doi: 10.1007/BF01076316. |
[35] |
R. Roussarie and F. O. O. Wagener, A study of the Bogdanov-Takens bifurcation,, Resenhas IME-USP, 2 (1995), 1.
|
[36] |
H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes,, Nachr. Ak. Wiss. Gött., 5 (1970), 67.
|
[37] |
H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus,, Dynamical Systems, 38 (1975), 598.
|
[38] |
M. B. Sevryuk, KAM tori: Persistence and smoothness,, Nonlinearity, 21 (2008).
doi: 10.1088/0951-7715/21/10/T01. |
[39] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Univ. Press, (1970).
|
[40] |
F. Takens, Forced oscillations and bifurcations,, Communications of the Mathematical Institute 3, (1974), 1.
|
[41] |
F. O. O. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dynamical Systems, 18 (2003), 159.
doi: 10.1080/1468936031000117857. |
[42] |
F. O. O. Wagener, On the quasi-periodic $d$-fold degenerate bifurcation,, Journal of Differential Equations, 216 (2005), 261.
doi: 10.1016/j.jde.2005.06.013. |
[43] |
J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, Journal of Differential Equations, 235 (2007), 609.
doi: 10.1016/j.jde.2006.12.001. |
[44] |
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I,, Communications in Pure and Applied Mathematics, 28 (1975), 91.
|
[45] |
D. Zhang and J. Xu, Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition,, Journal of Mathematical Analysis and Applications, 323 (2006), 293.
doi: 10.1016/j.jmaa.2005.10.029. |
[46] |
D. Zhang and J. Xu, On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Discrete and Continuous Dynamical Systems, 16 (2006), 635.
doi: 10.3934/dcds.2006.16.635. |
[47] |
D. Zhang and J. Xu, Invariant tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Nonlinear Analysis, 67 (2007), 2240.
doi: 10.1016/j.na.2006.09.012. |
[48] |
D. Zhang and J. Xu, Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition,, Acta Mathematica Sinica-English Series, 24 (2008), 1625.
doi: 10.1007/s10114-008-6180-x. |
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