KAM tori: Persistence and smoothness

Nonlinearity

Volumn 21, Issue 10, 2008, Pages

  Sevryuk, Mikhail B ^a  

^a SEMENOV INSTITUTE OF CHEMICAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 56349135982     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/21/10/T01     Document Type: Article

References (68)

1. Adams J F 1969 Lectures on Lie Groups (New York: Benjamin)
2. Albrecht J 2007 On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations Regular Chaotic Dyn. 12 281-320
3. Arnold V I 1964 On the instability of dynamical systems with many degrees of freedom Sov. Math.-Dokl. 5 581-5
4. Arnold V I 1972 Lectures on bifurcations and versal families Russ. Math. Surv. 27 (5) 54-123
5. Arnold V I, Gusein-Zade S M and Varchenko A N 1985 Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts (Monogr. Math. vol 82) (Boston, MA: Birkhäuser)
6. Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical Aspects of Classical and Celestial Mechanics (Encycl. Math. Sci. vol 3) 3rd edn revised and enlarged (Berlin: Springer)
7. Bogoyavlenskii O I 1998 Conformal symmetries of dynamical systems and Poincaré 1892 concept of iso-energetic non-degeneracy C. R. Acad. Sci. Paris Sér. I Math. 326 213-8
8. Bogoyavlenskii O I 1998 Necessary and sufficient conditions for two Poincaré concepts of nondegeneracy of integrable Hamiltonian systems Dokl. Math. 57 377-9
9. Broer H W, Hoo J and Naudot V 2007 Normal linear stability of quasi-periodic tori J. Diff. Eqns 232 355-418
10. Broer H W and Huitema G B 1995 Unfoldings of quasi-periodic tori in reversible systems J. Dyn. Diff. Eqns 7 191-212
11. Broer H W, Huitema G B and Sevryuk M B 1996 Families of quasi-periodic motions in dynamical systems depending on parameters Nonlinear Dynamical Systems and Chaos (Progress in Nonlinear Differential Equations and Their Applications vol 19) ed H W Broer et al (Basel: Birkhäuser) pp 171-211
12. Broer H W, Huitema G B and Sevryuk M B 1996 Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos (Lecture Notes in Mathematics vol 1645) (Berlin: Springer)
13. Broer H W, Huitema G B and Takens F 1990 Unfoldings of quasi-periodic tori Mem. Am. Math. Soc. 83 (421) 1-81
14. Broer H W and Sevryuk M B 2008 KAM theory: quasi-periodicity in dynamical systems Handbook of Dynamical Systems vol 3 ed H W Broer et al (Amsterdam: North-Holland) at press
15. Bunimovich L A 2008 Relative volume of Kolmogorov-Arnold-Moser tori and uniform distribution, stickiness and nonstickiness in Hamiltonian systems Nonlinearity 21 T13-7
16. Cao R and You J 2007 Diophantine vectors in analytic submanifolds of Euclidean spaces Sci. China Ser. A 50 1334-8
17. Chierchia L and Qian D 2004 Moser's theorem for lower dimensional tori J. Diff. Eqns 206 55-93
18. Chow S-N, Li Y and Yi Y 2002 Persistence of invariant tori on submanifolds in Hamiltonian systems J. Nonlinear Sci. 12 585-617
19. Cong F, Hong J and Han Y 2007 Near-invariant tori on exponentially long time for Poisson systems J. Math. Anal. Appl. 334 59-68
20. Cong F, Küpper T, Li Y and You J 2000 KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems J. Nonlinear Sci. 10 49-68
21. Cong F, Li Y and Huang M 1996 Invariant tori for nearly twist mappings with intersection property Northeast. Math. J. 12 280-98
22. de la Llave R 2001 A tutorial on KAM theory Smooth Ergodic Theory and Its Applications (Proc. Symp. in Pure Mathematics vol 69) ed A B Katok et al (Providence, RI: American Mathematical Society) pp 175-292
23. Féjoz J 2004 Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman) Ergod. Theory Dyn. Syst. 24 1521-82
24. Herman M R 1983 Sur les courbes invariantes par les diffé omorphismes de l'anneau (vol 1) Astérisque 103-104 1-221
25. Herman M R 1986 Sur les courbes invariantes par les diffé omorphismes de l'anneau (vol 2) Astérisque 144 1-248
26. Herman M R 1988 Existence et non existence de tores invariants par des difféomorphismes symplectiques Séminaire: quations aux Dérivées Partielles vol 1987/88 (Palaiseau: cole Polytechnique, Centre de Mathématiques) exp 14
27. Herman M R 1989 Inégalités 'a priori' pour des tores lagrangiens invariants par des difféomorphismes symplectiques Publ. Math. Inst. Hautes Etud. Sci. 70 47-101
28. Hoo J 2005 Quasi-periodic bifurcations in a strong resonance: Combination tones in gyroscopic stabilisation PhD Thesis University of Groningen
29. Huang Q, Cong F and Li Y 2000 Persistence of hyperbolic invariant tori for Hamiltonian systems J. Diff. Eqns 164 355-79
30. Huang Q, Cong F and Li Y 2001 Persistence of elliptic invariant tori for Hamiltonian systems Nonlinear Anal. 45 241-60
31. Huitema G B 1988 Unfoldings of quasi-periodic tori PhD Thesis University of Groningen
32. Kleinbock D 2008 An extension of quantitative nondivergence and applications to Diophantine exponents Trans. Am. Math. Soc. at press
33. Kubichka A A and Parasyuk I O 2001 Bifurcation of a Whitney-smooth family of coisotropic invariant tori of a Hamiltonian system under a small deformation of the symplectic structure Ukr. Math. J. 53 701-18
34. Li Y and Yi Y 2002 Persistence of invariant tori in generalized Hamiltonian systems Ergod. Theory Dyn. Syst. 22 1233-61
35. Li Y and Yi Y 2003 A quasi-periodic Poincaré's theorem Math. Ann. 326 649-90
36. Li Y and Yi Y 2005 Persistence of hyperbolic tori in Hamiltonian systems J. Diff. Eqns 208 344-87
37. Li Y and Yi Y 2006 Nekhoroshev and KAM stabilities in generalized Hamiltonian systems J. Dyn. Diff. Eqns 18 577-614
38. Liu B, Zhu W and Han Y 2006 Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems J. Math. Anal. Appl. 322 251-75
39. Liu Z 2005 Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems Nonlinear Anal. 61 1319-42
40. Liu Z, Yihe D and Huang Q 2005 Persistence of hyperbolic tori in generalized Hamiltonian systems Northeast. Math. J. 21 447-64
41. Loveikin Y V and Parasyuk I O 2005 Theorem on a perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications Nonlinear Oscil. 8 487-512
42. Loveikin Y V and Parasyuk I O 2006 Bifurcation of coisotropic invariant tori under locally Hamiltonian perturbations of integrable systems and a nondegenerate deformation of the symplectic structure Nonlinear Oscil. 9 215-25
43. Loveikin Y V and Parasyuk I O 2007 Invariant tori of locally Hamiltonian systems close to conditionally integrable ones Ukr. Math. J. 59 70-99
44. Moser J 1962 On invariant curves of area-preserving mappings of an annulus Nachr. Akad. Wiss. Gött. II Math.-Phys. Kl. 1 1-20
45. Moser J 1966 On the theory of quasiperiodic motions SIAM Rev. 8 145-72
46. Moser J 1967 Convergent series expansions for quasi-periodic motions Math. Ann. 169 136-76
47. Parasyuk I O 1984 On the preservation of multidimensional invariant tori of Hamiltonian systems Ukr. Math. J. 36 380-5
48. Parasyuk I O 1985 Multidimensional invariant tori of locally Hamiltonian systems and Kolmogorov-Arnold-Moser theory Usp. Mat. Nauk 40 (5) 217-8 (in Russian)
49. Parasyuk I O 1985 Coisotropic invariant tori of locally Hamiltonian systems Asymptotic Integration of Differential Equations ed Y A Mitropolskii (Kiev: Inst. Mat. Akad. Nauk USSR) pp 129-33 (in Russian)
50. Parasyuk I O 1986 On the preservation of coisotropic invariant tori of locally Hamiltonian systems Some Problems in the Theory of Asymptotic Methods in Nonlinear Mechanics ed Y A Mitropolskii et al (Kiev: Inst. Mat. Akad. Nauk USSR) pp 150-4 (in Russian)
51. Parasyuk I O 1990 Coisotropic invariant tori of Hamiltonian systems in the quasiclassical theory of the motion of a conduction electron Ukr. Math. J. 42 308-12
52. Parasyuk I O 1993 Variables of the action-angle type on symplectic manifolds foliated into coisotropic tori Ukr. Math. J. 45 85-93
53. Rüssmann H 2001 Invariant tori in non-degenerate nearly integrable Hamiltonian systems Regular Chaotic Dyn. 6 119-204
54. Salamon D A 2004 The Kolmogorov-Arnold-Moser theorem Math. Phys. Electron. J. 10 paper 3
55. Sevryuk M B 1995 The iteration-approximation decoupling in the reversible KAM theory Chaos 5 552-65
56. Sevryuk M B 1995 KAM-stable Hamiltonians J. Dyn. Control Syst. 1 351-66
57. Sevryuk M B 1996 Invariant tori of Hamiltonian systems nondegenerate in the sense of Rüssmann Dokl. Math. 53 69-72
58. Sevryuk M B 1999 The lack-of-parameters problem in the KAM theory revisited Hamiltonian Systems with Three or More Degrees of Freedom (NATO ASI Series C: Math. Phys. Sci. vol 533) ed C Simó (Dordrecht: Kluwer) pp 568-72
59. Sevryuk M B 2003 The classical KAM theory at the dawn of the twenty-first century Moscow Math. J. 3 1113-44
60. Sevryuk M B 2006 Partial preservation of frequencies in KAM theory Nonlinearity 19 1099-140
61. Sevryuk M B 2007 Partial preservation of frequencies and Floquet exponents in KAM theory Proc. Steklov Inst. Math. 259 167-95
62. Sevryuk M B 2007 Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method Discrete Contin. Dyn. Syst. Ser. A 18 569-95
63. Treshchv D V 1991 The mechanism of destruction of resonant tori in Hamiltonian systems Math. USSR Sbornik 68 181-203
64. Wagener F 2003 A note on Gevrey regular KAM theory and the inverse approximation lemma Dyn. Syst. 18 159-63
65. Wang S, Zhang D and Xu J 2007 On the persistence of elliptic lower-dimensional tori in Hamiltonian systems under the first Melnikov condition and Rüssmann's non-degeneracy condition Nonlinear Anal. 66 1675-85
66. Xu J and You J 2001 Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition J. Math. Pures Appl. Ser. 9 80 1045-67
67. Yoccoz J-C 1992 Travaux de Herman sur les tores invariants (Séminaire Bourbaki vol 1991/92 exp 754) Astérisque 206 311-44
68. Zhu W, Huang Q and Liu B 2004 The persistence of invariant tori in nearly small twist mappings with intersection property Northeast. Math. J. 20 175-90