Normal linear stability of quasi-periodic tori

Journal of Differential Equations

Volumn 232, Issue 2, 2007, Pages 355-418

  Broer, H W ^a   Hoo, J ^a   Naudot, V ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

(Uni )versal unfolding; Hamiltonian and reversible Hopf bifurcation; kam theory; Lagrange top; Linear centralizer unfolding; Nearly integrable system; Normal linear part; Perturbation theory; Quasi periodic bifurcations; Quasi periodic stability

Indexed keywords

EID: 33750612100     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.08.022     Document Type: Article

References (103)

1. Abraham R., and Marsden J.E. Foundations of Mechanics. Advanced Book Program (1978), Benjamin-Cummings, Reading, MA
2. Arnol'd V.I. Proof of a theorem of A.N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspekhi Mat. Nauk 18 5 (113) (1963) 13-40
3. Arnol'd V.I. On matrices depending on parameters. Russian Math. Surveys 26 2 (1971) 29-43
4. Arnol'd V.I. Lectures on bifurcations and versal families. Russian Math. Surveys 27 5 (1972) 54-123
5. Arnol'd V.I. Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss. vol. 250 (1983), Springer-Verlag, Berlin
6. Arnol'd V.I. Reversible matrices. In: Sagdeev R.Z. (Ed). Nonlinear and Turbulent Processes in Physics, vol. 3 (1984), Harwood Academic, Chur, NY 1161-1174
7. Arnol'd V.I. Mathematical Methods of Classical Mechanics. Grad. Texts in Math. vol. 60 (1989), Springer-Verlag, Berlin
8. Arnol'd V.I., and Sevryuk M.B. Oscillations and bifurcations in reversible systems. In: Sagdeev R.Z. (Ed). Nonlinear Phenomena in Plasma Physics and Hydrodynamics (1986), Mir, Moscow 31-64
9. In: Arnol'd V.I. (Ed). Arnold's Problems (2004), Springer-Verlag/Phasis, Berlin/Moscow
10. Bibikov Yu.N. Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Math. vol. 702 (1979), Springer-Verlag, Berlin
11. Bochnak J., Coste M., and Roy M.-F. Real Algebraic Geometry. Ergeb. Math. Grenzgeb. (3) (Results in Mathematics and Related Areas (3)) vol. 36 (1998), Springer-Verlag, Berlin
12. Braaksma B.L.J., and Broer H.W. On a quasiperiodic Hopf bifurcation. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 2 (1987) 115-168
13. Braaksma B.L.J., Broer H.W., and Huitema G.B. Toward a quasi-periodic bifurcation theory. Mem. Amer. Math. Soc. 83 421 (1990) 83-175
14. Bredon G.E. Introduction to Compact Transformation Groups (1972), Academic Press, New York
15. Bridges T. Poisson structure of the reversible 1 : 1 resonance. Phys. D 112 (1998) 40-49
16. H.W. Broer, M.C. Ciocci, H. Hanßmann, The quasi-periodic reversible Hopf bifurcation, IJBF (2006), in press
17. H.W. Broer, M.C. Ciocci, A. Vanderbauwhede, Normal 1 : 1 resonance of invariant tori in reversible systems: A study in quasi-periodic bifurcation theory, in preparation
18. H.W. Broer, R.H. Cushman, F. Fasso, F. Takens, Geometry of KAM tori for nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, in press
19. H.W. Broer, H. Hanßmann, J. Hoo, The quasi-periodic Hamiltonian Hopf bifurcation, preprint, University of Groningen, 2004
20. H.W. Broer, H. Hanßmann, F.O.O. Wagener, On quasi-periodic bifurcation theory, in preparation
21. Broer H.W., Hanßmann H., and You J. Bifurcation of normally parabolic tori in Hamiltonian systems. Nonlinearity 18 (2005) 1735-1769
22. Broer H.W., Hanßmann H., and You J. Umbilical torus bifurcations in Hamiltonian systems. J. Differential Equations 222 (2006) 233-262
23. Broer H.W., and Huitema G.B. Unfoldings of quasi-periodic tori in reversible systems. J. Dynam. Differential Equations 7 1 (1995) 191-212
24. Broer H.W., Huitema G.B., and Sevryuk M.B. Families of quasi-periodic motions in dynamical systems depending on parameters. In: Broer H.W., van Gils S.A., Hoveijn I., and Takens F. (Eds). Nonlinear Dynamical Systems and Chaos. Groningen, 1995. Progr. Nonlinear Differential Equations Appl. vol. 19 (1996), Birkhäuser, Basel 171-211
25. Broer H.W., Huitema G.B., and Sevryuk M.B. Quasi-periodic Motions in Families of Dynamical Systems. Lecture Notes in Math. vol. 1645 (1996), Springer-Verlag, Berlin
26. Broer H.W., Huitema G.B., and Takens F. Unfoldings of quasi-periodic tori. Mem. Amer. Math. Soc. 83 421 (1990) 1-82
27. H.W. Broer, M.B. Sevryuk, KAM theory: Quasi-periodicity in dynamical systems, in: H.W. Broer, B. Hasselblatt, F. Takens (Eds.), Handbook of Dynamical Systems, vol. 3, North-Holland, in press
28. Bruno A.D. The Restricted 3-Body Problem: Plane Periodic Orbits. de Gruyter Exp. Math. vol. 17 (1994), de Gruyter, Berlin
29. Burgoyne N., and Cushman R.H. Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues. Celestial Mech. 8 (1973/1974) 435-443
30. Chevalley C. Theory of Lie Groups, vol. I (1946), Princeton Univ. Press, Princeton
31. M.C. Ciocci, Bifurcation of periodic orbits and persistence of quasi-periodic orbits in families of reversible systems, PhD thesis, Ghent University, 2003
32. Ciocci M.C. Persistence of quasi-periodic orbits in families of reversible systems with a 1 : 1 resonance. In: Dumortier F., et al. (Ed). Proceedings Equadiff 03. Hasselt, 2003 (2005), World Scientific, Singapore 720-725
33. Contopoulos G., and Mertzanides C. Inner Lindblad resonance in galaxies. Nonlinear theory. II. Bars. Astron. Astrophys. 61 4 (1977) 477-485
34. Cushman R.H., and Bates L.M. Global Aspects of Classical Integrable Systems (1997), Birkhäuser, Basel
35. Cushman R.H., and van der Meer J.-C. The Hamiltonian Hopf bifurcation in the Lagrange top. Géométrie Symplectique et Mécanique. La Grande Motte, 1988. Lecture Notes in Math. vol. 1416 (1990), Springer-Verlag, Berlin 26-38
36. Debnath L., and Mikusiński P. Introduction to Hilbert Spaces with Applications (1990), Academic Press, New York
37. Dodson M.M., Pöschel J., Rynne B.P., and Vickers J.A.G. The Hausdorff dimension of small divisors for lower-dimensional KAM-tori. Proc. R. Soc. Lond. Ser. A 439 1906 (1992) 359-371
38. Ferrer S., Hanßmann H., Palacián J., and Yanguas P. On perturbed oscillators in 1 : 1 : 1 resonance: The case of axially symmetric cubic potentials. J. Geom. Phys. 40 3-4 (2002) 320-369
39. Galin D.M. Versal deformations of linear Hamiltonian systems. Trudy Sem. Petrovsk. (1) (1975) 63-74
40. Gibson C.G. Singular Points of Smooth Mappings. Advanced Publishing Program. Res. Notes Math. vol. 25 (1979), Pitman
41. A.A. Glutsyuk, Upper bounds of topology of complex polynomials in two variables, Moscow Math. J. (2006), in press
42. Goldstein H. Classical Mechanics (1951), Addison-Wesley, Cambridge, MA
43. Hanßmann H. The quasi-periodic center-saddle bifurcation. J. Differential Equations 142 (1998) 305-370
44. Hanßmann H. Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems-Results and Examples, 2004. Lecture Notes in Math. vol. 1893 (2006), Springer-Verlag
45. Hanßmann H. On Hamiltonian bifurcations of invariant tori with a Floquet multiplier -1. Dyn. Syst. 21 2 (2006) 115-145
46. Hanßmann H., and van der Meer J.-C. On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family. J. Dynam. Differential Equations 14 3 (2002) 675-695
47. Hanßmann H., and van der Meer J.-C. Algebraic methodes for determining Hamiltonian Hopf bifurcations in three-degrees-of-freedom systems. J. Dynam. Differential Equations 17 3 (2005) 453-474
48. Hausdorff F. Set Theory. second ed. (1962), Chelsea, New York Translated from the German by J.R. Aumann, et al
49. Helgason S. Differential Geometry and Symmetric Spaces. Pure Appl. Math. vol. XII (1962), Academic Press
50. Hirsch M.W. Differential Topology (1976), Springer-Verlag, Berlin
51. J. Hoo, Matrices depending on parameters, with focus on some resonant Hamiltonian cases, Master's thesis, University of Groningen, 2001
52. J. Hoo, Quasi-periodic bifurcations in a strong resonance, combination tones in gyroscopic stabilization, PhD thesis, University of Groningen, 2005
53. Hoveijn I. Versal deformations and normal forms for reversible and Hamiltonian linear systems. J. Differential Equations 126 2 (1996) 408-442
54. Hoveijn I., Lamb J.S.W., and Roberts R.M. Normal forms and unfoldings of linear systems in eigenspaces of (anti-)automorphisms of order two. J. Differential Equations 190 1 (2003) 182-213
55. G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988
56. H.H. de Jong, Quasiperiodic breathers in systems of weakly coupled pendulums, PhD thesis, University of Groningen, 1999
57. Knobloch J., and Vanderbauwhede A. Hopf bifurcation at k-fold resonances in conservative systems. In: Broer H.W., van Gils S.A., Hoveijn I., and Takens F. (Eds). Nonlinear Dynamical Systems and Chaos. Groningen, 1995. Progr. Nonlinear Differential Equations Appl. vol. 19 (1996), Birkhäuser, Basel 155-170
58. J. Knobloch, A. Vanderbauwhede, Hopf bifurcation at k-fold resonances in reversible systems, preprint N, M16/95, Technische Universität Ilmenau, 1996
59. Kolmogorov A.N. On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR (N.S.) 98 (1954) 527-530
60. Lamb J.S.W., and Roberts J.A.G. Time-reversal symmetry in dynamical systems: A survey. Phys. D 112 (1998) 1-39
61. Landau L.D., and Lifshitz E.M. Mechanics. Course of Theoretical Phys. vol. 1 (1960), Pergamon, Elmsford, NY Translated from the Russian by J.B. Bell
62. Marsden J., and Weinstein A. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5 1 (1974) 121-130
63. Marsden J.E., and Scheurle J. Lagrangian reduction and the double spherical pendulum. Z. Angew. Math. Phys. 44 1 (1993) 17-43
64. van der Meer J.-C. The Hamiltonian Hopf bifurcation. Lecture Notes in Math. vol. 1160 (1985), Springer-Verlag, Berlin
65. van Meer J.C., Sanders J.A., and Vanderbauwhede A. Hamiltonian structure of the reversible nonsemisimple 1 : 1 resonance. In: Chossat P. (Ed). Dynamics, Bifurcations and Symmetry. Cargese, 1993 (1994), Kluwer, Dordrecht 221-240
66. Mel'nikov V.K. On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function. Dokl. Akad. Nauk SSSR 165 (1965) 1245-1248
67. Mel'nikov V.K. A certain family of conditionally periodic solutions of a Hamiltonian system. Dokl. Akad. Nauk SSSR 181 (1968) 546-549
68. Montgomery D., and Zippin L. Topological Transformation Groups (1955), Interscience, New York
69. Moser J.K. Combination tones for Duffing's equation. Comm. Pure Appl. Math. 18 (1965) 167-181
70. Moser J.K. On the theory of quasiperiodic motions. SIAM Rev. 8 2 (1966) 145-172
71. Moser J.K. Convergent series expansions for quasi-periodic motions. Math. Ann. 169 (1967) 136-176
72. Moser J.K. Remark on the paper "On invariant curves of area-preserving mappings of an annulus,". Regul. Chaotic Dyn. 6 (2001) 337-338
73. Moser J.K. Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics. Ann. of Math. Stud. vol. 77 (2001), Princeton Univ. Press, Princeton, NJ
74. Nekhorošev N.N. Action-angle variables, and their generalizations. Trudy Moskov. Mat. Obšč. 26 (1972) 181-198
75. J.R. Pacha, On the quasi-periodic Hamiltonian Andronov-Hopf bifurcation, PhD thesis, Universitat Politècnica de Catalunya, 2002
76. Pfenniger D. The 3D dynamics of barred galaxies. Astron. Astrophys. 134 2 (1984) 373-386
77. Pontrjagin L. Topological Groups. Princeton Math. Ser. vol. 2 (1939), Princeton Univ. Press, Princeton
78. Pöschel J. Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35 5 (1982) 653-696
79. Pöschel J. On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202 4 (1989) 559-608
80. Quispel G.R.W., and Sevryuk M.B. kam theorems for the product of two involutions of different types. Chaos 3 (1993) 757-769
81. Raţiu T., and van Moerbeke P. The Lagrange rigid body motion. Ann. Inst. Fourier (Grenoble) 32 1 (1982) 211-234
82. Roberts J.A.G., and Quispel G.R.W. Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216 2-3 (1992) 63-177
83. Rüssmann H. Kleine Nenner II. Bemerkungen zur Newtonschen Methode. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1972) 1-10
84. Rüssmann H. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6 2 (2001) 119-204
85. Sevryuk M.B. Reversible Systems. Lecture Notes in Math. vol. 1211 (1986), Springer-Verlag, Berlin
86. Sevryuk M.B. Lower-dimensional tori in reversible systems. Chaos 1 (1991) 160-167
87. Sevryuk M.B. Linear reversible systems and their versal deformations. J. Sov. Math. 60 2 (1992) 1663-1680
88. Sevryuk M.B. Some problems of KAM theory: Conditionally periodic motions in typical systems. Russian Math. Surveys 50 2 (1995) 341-353
89. Sevryuk M.B. The finite-dimensional reversible kam theory. Phys. D 112 (1998) 132-147
90. Sevryuk M.B. On the convergence of coordinate transformations in the kam procedure. Regul. Chaotic Dyn. 5 (2000) 181-188
91. Sevryuk M.B. The comment to problem 1970-1. In: Arnold V.I. (Ed). Arnold's Problems (2004), Springer-Verlag/Phasis, Berlin/Moscow 226-231
92. M.B. Sevryuk, Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method, Discrete Contin. Dyn. Syst. Ser. A (2006), in press
93. Sevryuk M.B. Partial preservation of frequencies in kam theory. Nonlinearity (2006) 1099-1140
94. Stein E.M. Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser. vol. 30 (1970), Princeton Univ. Press, Princeton
95. Varadarajan V.S. Lie Groups, Lie Algebras, and Their Representations. Grad. Texts in Math. vol. 102 (1984), Springer-Verlag, Berlin
96. Wagener F.O.O. A note on Gevrey regular KAM theory and the inverse approximation lemma. Dyn. Syst. 18 2 (2003) 159-163
97. F.O.O. Wagener, A parametrised version of Moser's modifying terms theorem, in preparation
98. Weinstein A. Lectures on Symplectic Manifolds (1977), American Mathematical Society, Providence, RI
99. Whitney H. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 1 (1934) 63-89
100. Whittaker E.T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (1944), Dover, New York
101. Williamson J. On the algebraic problem concerning the normal forms of linear dynamical systems. Amer. J. Math. 58 (1936) 141-163
102. Zehnder E. Generalized Implicit Function Theorems with applications to some small divisor problems, I. Comm. Pure Appl. Math. 28 (1975) 91-140
103. Zehnder E. Generalized Implicit Function Theorems with applications to some small divisor problems, II. Comm. Pure Appl. Math. 29 (1976) 49-111