Quasiperiodic dynamics in Hamiltonian 1 1/2 degree of freedom systems far from integrability

Journal of Differential Equations

Volumn 212, Issue 2, 2005, Pages 366-393

  Chow, Shui Nee ^a   van Noort, Martijn ^a   Yi, Yingfei ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Aubry Mather theory; KAM theory; Non integrable Hamiltonian systems

Indexed keywords

EID: 16844370861     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2005.01.008     Document Type: Article

References (33)

1. V.I. Arnol'd Small denominators and problems of stability of motion in classical and celestial mechanics Uspehi Mat. Nauk (18) 6 114 1963 91-192 (translated in Russian Math. Surveys)
2. V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, second ed., Springer, Berlin, 1989.
3. S. Aubry P.Y. Le Daeron The discrete Frenkel-Kontorova model and its extensions. I, Exact results for the ground-states Phys. D 8 3 1983 381-422
4. H.W. Broer, I. Hoveijn, M. van Noort, C. Simó, G. Vegter, The parametrically forced pendulum: A case study in 1 1/2 degree of freedom, JDDE 16(4), p. 897-947.
5. H.W. Broer G.B. Huitema M.B. Sevryuk Quasi-periodic motions in families of dynamical systems, Order amidst chaos Lecture Notes in Mathematics vol. 1645 1996 Springer Berlin
6. H.W. Broer, M. van Noort, C. Simó, Existence and measure of 2-quasiperiodicity in Hamiltonian one-and-a-half degree of freedom systems, Proceedings of Equadiff 2003, to appear.
7. H.W. Broer R. Roussarie C. Simó Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms Ergodic Theory Dynam. Systems 16 6 1996 1147-1172
8. A. Delshams P. Gutiérrez Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system J. Differential Equations 131 2 1996 277-303
9. R. Dieckerhoff E. Zehnder Boundedness of solutions via the twist-theorem Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 1 1987 79-95
10. Fassò F. Lie series method for vector fields and Hamiltonian perturbation theory Z. Angew. Math. Phys. 41 6 1990 843-864
11. G. Gallavotti G. Gentile V. Mastropietro A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems J. Math. Phys. 40 12 1999 6430-6472
12. M.R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, vol. 1, vol. 103-104 of Astérisque, Société Mathématique de France, 1983.
13. M.R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, vol. 2, vol. 144 of Astérisque, Société Mathématique de France, 1986.
14. M. Jiang On a theorem by Mather and Aubry-Mather sets for planar Hamiltonian systems Sci. China Ser. A 42 11 1999 1121-1128
15. À. Jorba J. Villanueva On the persistence of lower-dimensional invariant tori under quasi-periodic perturbations J. Nonlinear Sci. 7 5 1997 427-473
16. S. Laederich M. Levi Invariant curves and time-dependent potentials Ergod. Theory Dynam. Systems 11 2 1991 365-378
17. M. Levi KAM theory for particles in periodic potentials Ergod. Theory Dynam. Systems 10 4 1990 777-785
18. M. Levi A period-adding phenomenon SIAM J. Appl. Math. 50 4 1990 943-955
19. M. Levi Quasiperiodic motions in superquadratic time-periodic potentials Comm. Math. Phys. 143 1 1991 43-83
20. M. Levi E. Zehnder Boundedness of solutions for quasiperiodic potentials SIAM J. Math. Anal. 26 5 1995 1233-1256
21. J.E. Littlewood Unbounded solutions of an equation y ̈ + g(y) = p(t), with p(t) periodic and bounded and g(y)/y → ∞ as y → ±∞ J. London Math. Soc. 41 1966 491-507
22. B. Liu Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem J. Differential Equations 79 2 1989 304-315
23. J.N. Mather Existence of quasiperiodic orbits for twist homeomorphisms of the annulus Topology 21 4 1982 457-467
24. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. II (1962) 1-20.
25. J. Moser Recent developments in the theory of Hamiltonian systems SIAM Rev. 28 4 1986 459-485
26. J. Moser, Minimal foliations on a torus, in: M. Giaquinta (Ed.), Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Mathematics, vol. 1365, Springer, Berlin, 1989, pp. 62-99.
27. J. Moser Quasi-periodic solutions of nonlinear elliptic partial differential equations Bol. Soc. Brasil. Mat. (NS) 20 1 1989 29-45
28. A.I. Neishtadt The separation of motions in systems with rapidly rotating phase J. Appl. Math. Mech. 48 2 1984 133-139
29. J. Pöschel Integrability of Hamiltonian systems on Cantor sets Comm. Pure Appl. Math. 35 5 1982 653-696
30. H. Rüssmann, Kleine Nenner, I, Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1970) 67-105.
31. H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, in: J. Palis Jr. (Ed.), Geometric Dynamics, Lecture Notes in Mathematics, vol. 1007, Springer, Berlin, 1983, pp. 677-719.
32. C. Simó C. Valls A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters Nonlinearity 14 6 2001 1707-1760
33. J. You Invariant tori and Lagrange stability of pendulum-type equations J. Differential Equations 85 1990 54-65