# American Institute of Mathematical Sciences

December  2010, 3(4): 719-768. doi: 10.3934/dcdss.2010.3.719

## A parametrised version of Moser's modifying terms theorem

 1 CeNDEF, Dept. of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, Netherlands

Received  April 2009 Revised  May 2010 Published  August 2010

A sharpened version of Moser's 'modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is 'structural' and can be applied to dissipative, Hamiltonian, and symmetric vector fields; moreover, we give variants of the result for real analytic, Gevrey regular ultradifferentiable and finitely differentiable vector fields. In the first two cases, the conjugacy constructed in the theorem is shown to be Gevrey smooth in the sense of Whitney on the set of parameters that satisfy a "Diophantine'' non-resonance condition.
Citation: Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719
##### References:
 [1] V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer, New York, 1988, Second Edition. [2] V. I. Arnol'd, "Mathematical Methods Of Classical Mechanics," Springer, New York, 1989, Second Edition. [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), 23-35, Translation in: Selecta Math. Sovietica, 1 (1981), 373-388. [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), 37-65; Translation in: Selecta Math. Sovietica, 1 (1981), 389-421. [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), 155-184. [6] H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. 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), 2605-2623. doi: 10.1142/S021812740701866X. [8] H. W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems, Nonlinearity, 18 (2005), 1735-1769. 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), 355-418. 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, vol. 1645, Springer, Berlin, 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, vol. 83, no. 421, AMS, 1990. [12] H. W. Broer, R. Roussarie and C. Simó, On the Bogdanov-Takens bifurcation for planar diffeomorphisms, Proceedings of Equadiff '91 (C. Perelló, C. Simó and J. Solà-Morales, eds.), World Sci. Publ., River Edge, NJ, 1993, 81-92. [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), 1147-1172. [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), 69-85. [15] A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes, Publications Mathématiques de l'I.H.E.S., 61 (1985), 67-127. [16] M. C. Ciocci, "Bifurcation of Periodic Orbits and Persistence of Quasi Periodic Orbits in Families of Reversible Systems," Ph.D. thesis, Rijksuniversiteit Gent, 2003; available at http://lib.ugent.be. [17] H. H. de Jong, "Quasi-periodic Breathers in Systems of Weakly Coupled Pendulums: Applications of KAM Theory," Ph.D. thesis, Rijksuniversiteit Groningen, 1999. [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), 863-906. 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. [20] M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities," Graduate Texts in Mathematics, vol. 14, Springer, New York, 1973. [21] H. Hanßmann, The quasi-periodic centre-saddle bifurcation, Journal of Differential Equations, 142 (1998), 305-370. 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, vol. 1893, Springer, Berlin, 2007. [23] G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, Cambridge, 1952, Second Edition. [24] J. Hoo, "Quasi-periodic Bifurcations in a Strong Resonance: Combination Tones in Gyroscopic Stabilisation," Ph.D. thesis, Rijksuniversiteit Groningen, 2005. [25] L. Hörmander, "An Introduction to Complex Analysis in Several Variables," North-Holland, Amsterdam, 1990, Third edition. [26] L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Springer, Berlin, 1990, Second edition. [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. [28] J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176. doi: 10.1007/BF01399536. [29] V. Poénaru, "Analyse Différentielle," Lecture Notes in Mathematics, vol. 371, Springer, Berlin, 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), 223-248. 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), 249-279. [32] G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458. [33] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications of Pure and Applied Mathematics, 35 (1982), 653-696. doi: 10.1002/cpa.3160350504. [34] A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space, Functional Analysis and Applications, 3 (1969), 303-306. doi: 10.1007/BF01076316. [35] R. Roussarie and F. O. O. Wagener, A study of the Bogdanov-Takens bifurcation, Resenhas IME-USP, 2 (1995), 1-25. [36] H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Ak. Wiss. Gött., II Math. Phys. Kl., 5 (1970), 67-105. [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, Theory and Applications (J. Moser, ed.), Lecture Notes in Physics, vol. 38, Springer, Berlin, (1975), 598-624. [38] M. B. Sevryuk, KAM tori: Persistence and smoothness, Nonlinearity, 21 (2008), T177-T185. doi: 10.1088/0951-7715/21/10/T01. [39] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Univ. Press, Princeton, NJ, 1970. [40] F. Takens, Forced oscillations and bifurcations, Communications of the Mathematical Institute 3, Rijksuniversiteit Utrecht, 1974. Reprinted in: Global Analysis of Dynamical Systems (H.W. Broer, B. Krauskopf and G. Vegter, eds.), Inst. Phys. Publ., Bristol, 2001, pp. 1-61. [41] F. O. O. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dynamical Systems, 18 (2003), 159-163. doi: 10.1080/1468936031000117857. [42] F. O. O. Wagener, On the quasi-periodic $d$-fold degenerate bifurcation, Journal of Differential Equations, 216 (2005), 261-281. 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), 609-622. 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), 91-140. [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), 293-312. 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), 635-655. 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), 2240-2257. 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), 1625-1636. doi: 10.1007/s10114-008-6180-x.

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##### References:
 [1] V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer, New York, 1988, Second Edition. [2] V. I. Arnol'd, "Mathematical Methods Of Classical Mechanics," Springer, New York, 1989, Second Edition. [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), 23-35, Translation in: Selecta Math. Sovietica, 1 (1981), 373-388. [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), 37-65; Translation in: Selecta Math. Sovietica, 1 (1981), 389-421. [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), 155-184. [6] H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. 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), 2605-2623. doi: 10.1142/S021812740701866X. [8] H. W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems, Nonlinearity, 18 (2005), 1735-1769. 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), 355-418. 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, vol. 1645, Springer, Berlin, 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, vol. 83, no. 421, AMS, 1990. [12] H. W. Broer, R. Roussarie and C. Simó, On the Bogdanov-Takens bifurcation for planar diffeomorphisms, Proceedings of Equadiff '91 (C. Perelló, C. Simó and J. Solà-Morales, eds.), World Sci. Publ., River Edge, NJ, 1993, 81-92. [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), 1147-1172. [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), 69-85. [15] A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes, Publications Mathématiques de l'I.H.E.S., 61 (1985), 67-127. [16] M. C. Ciocci, "Bifurcation of Periodic Orbits and Persistence of Quasi Periodic Orbits in Families of Reversible Systems," Ph.D. thesis, Rijksuniversiteit Gent, 2003; available at http://lib.ugent.be. [17] H. H. de Jong, "Quasi-periodic Breathers in Systems of Weakly Coupled Pendulums: Applications of KAM Theory," Ph.D. thesis, Rijksuniversiteit Groningen, 1999. [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), 863-906. 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. [20] M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities," Graduate Texts in Mathematics, vol. 14, Springer, New York, 1973. [21] H. Hanßmann, The quasi-periodic centre-saddle bifurcation, Journal of Differential Equations, 142 (1998), 305-370. 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, vol. 1893, Springer, Berlin, 2007. [23] G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, Cambridge, 1952, Second Edition. [24] J. Hoo, "Quasi-periodic Bifurcations in a Strong Resonance: Combination Tones in Gyroscopic Stabilisation," Ph.D. thesis, Rijksuniversiteit Groningen, 2005. [25] L. Hörmander, "An Introduction to Complex Analysis in Several Variables," North-Holland, Amsterdam, 1990, Third edition. [26] L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Springer, Berlin, 1990, Second edition. [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. [28] J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176. doi: 10.1007/BF01399536. [29] V. Poénaru, "Analyse Différentielle," Lecture Notes in Mathematics, vol. 371, Springer, Berlin, 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), 223-248. 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), 249-279. [32] G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458. [33] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications of Pure and Applied Mathematics, 35 (1982), 653-696. doi: 10.1002/cpa.3160350504. [34] A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space, Functional Analysis and Applications, 3 (1969), 303-306. doi: 10.1007/BF01076316. [35] R. Roussarie and F. O. O. Wagener, A study of the Bogdanov-Takens bifurcation, Resenhas IME-USP, 2 (1995), 1-25. [36] H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Ak. Wiss. Gött., II Math. Phys. Kl., 5 (1970), 67-105. [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, Theory and Applications (J. Moser, ed.), Lecture Notes in Physics, vol. 38, Springer, Berlin, (1975), 598-624. [38] M. B. Sevryuk, KAM tori: Persistence and smoothness, Nonlinearity, 21 (2008), T177-T185. doi: 10.1088/0951-7715/21/10/T01. [39] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Univ. Press, Princeton, NJ, 1970. [40] F. Takens, Forced oscillations and bifurcations, Communications of the Mathematical Institute 3, Rijksuniversiteit Utrecht, 1974. Reprinted in: Global Analysis of Dynamical Systems (H.W. Broer, B. Krauskopf and G. Vegter, eds.), Inst. Phys. Publ., Bristol, 2001, pp. 1-61. [41] F. O. O. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dynamical Systems, 18 (2003), 159-163. doi: 10.1080/1468936031000117857. [42] F. O. O. Wagener, On the quasi-periodic $d$-fold degenerate bifurcation, Journal of Differential Equations, 216 (2005), 261-281. 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), 609-622. 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), 91-140. [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), 293-312. 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), 635-655. 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), 2240-2257. 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), 1625-1636. doi: 10.1007/s10114-008-6180-x.
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