A parametrised version of moser's modifying terms theorem

Discrete and Continuous Dynamical Systems - Series S

Volumn 3, Issue 4, 2010, Pages 719-768

  Wagener, Florian ^a  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

Modifying terms; Normal internal resonances; Quasi periodic tori

Indexed keywords

EID: 79952018443     PISSN: 19371632     EISSN: 19371179     Source Type: Journal    
DOI: 10.3934/dcdss.2010.3.719     Document Type: Conference Paper

References (50)

1. V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer, New York, 1988, Second Edition.
2. 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
3. Translation in: Selecta Math. Sovietica, 1 (1981), 373-388.
4. -, 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;
5. Translation in: Selecta Math. Sovietica, 1 (1981), 389-421.
6. J. Bonet, R. W. Braun, R. Meise and B. A. Taylor, Whitney's extension theorem for nonquasi-analytic classes of ultradifferentiable functions, Studia Mathematica, 99 (1991), 155-184.
7. H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318.
8. 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. (Pubitemid 47479146)
9. H. W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems, Nonlinearity, 18 (2005), 1735-1769. (Pubitemid 40900811)
10. H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, Journal of Differential Equations, 232 (2007), 355-418. (Pubitemid 44688961)
11. 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.
12. 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.
13. H. W. Broer, R. Roussarie and C. Siḿo, On the Bogdanov-Takens bifurcation for planar diffeomorphisms, Proceedings of Equadiff '91 (C. Pereló, C. Siḿo and J. Sol̀a-Morales, eds.), World Sci. Publ., River Edge, NJ, 1993, 81-92.
14. -, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms, Ergodic Theory and Dynamical Systems, 16 (1996), 1147-1172.
15. T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomor-phisms of (C, 0), Bulletin de la Socíet'e Math'ematique de France, 128 (2000), 69-85.
16. A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes, Publications Math́ematiques de l'I.H.E.S., 61 (1985), 67-127.
17. 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.
18. H. H. de Jong, "Quasi-periodic Breathers in Systems of Weakly Coupled Pendulums: Applications of KAM Theory," Ph.D. thesis, Rijksuniversiteit Groningen, 1999.
19. 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.
20. A. Gonźalez-Enŕiquez, A. Haro and R. de la Llave, Translated torus theory with parameters and applications, In preparation.
21. M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities," Graduate Texts in Mathematics, vol. 14, Springer, New York, 1973.
22. H. Hanßmann, The quasi-periodic centre-saddle bifurcation, Journal of Differential Equations, 142 (1998), 305-370. (Pubitemid 128345979)
23. -, "Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples," Lecture Notes in Mathematics, vol. 1893, Springer, Berlin, 2007.
24. G. H. Hardy, J. E. Littlewood and G. P'olya, "Inequalities," Cambridge University Press, Cambridge, 1952, Second Edition.
25. J. Hoo, "Quasi-periodic Bifurcations in a Strong Resonance: Combination Tones in Gyroscopic Stabilisation," Ph.D. thesis, Rijksuniversiteit Groningen, 2005.
26. L. Ḧormander, "An Introduction to Complex Analysis in Several Variables," North-Holland, Amsterdam, 1990, Third edition.
27. -, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Springer, Berlin, 1990, Second edition.
28. 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.
29. J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.
30. V. Póenaru, "Analyse Diff'erentielle," Lecture Notes in Mathematics, vol. 371, Springer, Berlin, 1974.
31. G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I: Birkhoff normal forms, Annales de l'Institut Henri Poincaŕe, 1 (2000), 223-248.
32. -, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms, Annales de l'Institut Henri Poincaŕe, 1 (2000), 249-279.
33. -, KAM theorem for Gevrey Hamiltonians, Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786.
34. J. P̈oschel, Integrability of Hamiltonian systems on Cantor sets, Communications of Pure and Applied Mathematics, 35 (1982), 653-696.
35. A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space, Functional Analysis and Applications, 3 (1969), 303-306.
36. R. Roussarie and F. O. O. Wagener, A study of the Bogdanov-Takens bifurcation, Resenhas IME-USP, 2 (1995), 1-25.
37. H. Rüssmann, Kleine Nenner I: Uber invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Ak. Wiss. G̈ott., II Math. Phys. Kl., 5 (1970), 67-105.
38. -, 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.
39. M. B. Sevryuk, KAM tori: Persistence and smoothness, Nonlinearity, 21 (2008), T177-T185.
40. E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Univ. Press, Princeton, NJ, 1970.
41. F. Takens, Forced oscillations and bifurcations, Communications of the Mathematical Institute 3, Rijksuniversiteit Utrecht, 1974.
42. Reprinted in: Global Analysis of Dynamical Systems (H.W. Broer, B. Krauskopf and G. Vegter, eds.), Inst. Phys. Publ., Bristol, 2001, pp. 1-61.
43. F. O. O. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dynamical Systems, 18 (2003), 159-163.
44. -, On the quasi-periodic d-fold degenerate bifurcation, Journal of Differential Equations, 216 (2005), 261-281.
45. J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamilton-ian systems under R̈ussmann's non-degeneracy condition, Journal of Differential Equations, 235 (2007), 609-622.
46. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I, Communications in Pure and Applied Mathematics, 28 (1975), 91-140.
47. D. Zhang and J. Xu, Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamil-tonian systems under R̈ussmann's non-degeneracy condition, Journal of Mathematical Analysis and Applications, 323 (2006), 293-312. (Pubitemid 44276152)
48. -, 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.
49. -, Invariant tori for Gevrey-smooth Hamiltonian systems under R̈ussmann's non-degeneracy condition, Nonlinear Analysis, 67 (2007), 2240-2257. (Pubitemid 46659801)
50. -, Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition, Acta Mathematica Sinica-English Series, 24 (2008), 1625-1636.