Splitting of separatrices in the resonances of nearly integrable hamiltonian systems of one and a half degrees of freedom

Discrete and Continuous Dynamical Systems- Series A

Volumn 33, Issue 7, 2013, Pages 2829-2859

  Guardia, Marcel ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Complex matching; Exponentially small splitting of separatrices; Melnikov method; Nearly integrable Hamiltonian systems

Indexed keywords

EID: 84872121501     PISSN: 10780947     EISSN: 15535231     Source Type: Journal    
DOI: 10.3934/dcds.2013.33.2829     Document Type: Article

References (36)

1. V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Dynamical Systems III, " 3 of Encyclopaedia Math. Sci., Springer, Berlin, 1988.
2. V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations, Russian Math. Surveys, 18 (1963), 9-36.
3. I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems, Nonlinearity, 19 (2006), 1415-1445. (Pubitemid 43752412)
4. I. Baldomá and E. Fontich, Exponentially small splitting of invariant manifolds of parabolic points, Mem. Amer. Math. Soc., 167 (2004), x-83.
5. I. Baldomá and E. Fontich, Exponentially small splitting of separatrices in a weakly hyperbolic case, J. dieffrential Equations, 210 (2005), 106-134. (Pubitemid 40140091)
6. I. Baldomá, E. Fontich, M. Guárdia and T. M. Seara, Exponentially small splitting of separatrices beyond melnikov analysis: Rigorous results, preprint, arXiv:1201.5152, 2011.
7. L. Chierchia and G. Gallavotti, Drift and dińusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144 pp.
8. A. Delshams, V. Gelfreich, Á. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1997), 35-71. (Pubitemid 127358543)
9. A. Delshams, P. Gutiérrez and T. M. Seara, Exponentially small splitting for whiskered tori in Hamiltonian sysems: Flow-box coordinates and upper bounds, Discrete Contin. Dyn. Syst., 11 (2004), 785-826. (Pubitemid 40195126)
10. A. Delshams and T. M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys., 150 (1992), 433-463.
11. A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3 (1997), 40 pp. (electronic).
12. E. Fontich, Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations, Nonlinear Anal., 20 (1993), 733-744.
13. E. Fontich, Rapidly forced planar vector élds and splitting of separatrices, J. dieffrential Equations, 119 (1995), 310-335.
14. V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, in "Seminar on Dynamical Systems (St. Petersburg, 1991)" 12 of Progr. Nonlinear dieffrential Equations Appl., 47-67. Birkhäuser, Basel, (1994).
15. V. G. Gelfreich, Melnikov method and exponentially small splitting of separatrices, Phys. D, 101 (1997), 227-248. (Pubitemid 127701364)
16. V. G. Gelfreich, Reference systems for splittings of separatrices, Nonlinearity, 10 (1997), 175-193.
17. V. G. Gelfreich, Separatrix splitting for a high-frequency perturbation of the pendulum, Russ. J. Math. Phys., 7 (2000), 48-71.
18. G. Gallavotti, G. Gentile and V. Mastropietro, Separatrix splitting for systems with three time scales, Comm. Math. Phys., 202 (1999), 197-236.
19. J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, " Springer-Verlag, 1983.
20. M. Guardia, C. Olivé and T. Seara, Exponentially small splitting for the pendulum: A classical problem revisited, J. Nonlinear Sci., 20 (2010), 595-685.
21. M. Guardia and T. M. Seara, Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation, Nonlinearity, 25 (2012), 1367-1412.
22. P. Holmes, J. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, in "Hamiltonian Dynamical Systems" 81 of Contemp. Math. (1988).
23. A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.
24. P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163 (2003), viii+145.
25. V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57.
26. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1962), 1-20.
27. A. I. Neíshtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., 48 (1984), 197-204.
28. C. Olivé, "Cálcul de Léscissińo de Separatrius Usant Técniques de Matching Complex I Ressurgéncia Aplicades a Léquacińo de Hamilton-Jacobi, " Ph.D thesis, Universitat Politécnica de Catalunya, 2006.
29. C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation, in "Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002)" 53 (2003), 1185-1235.
30. H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-270.
31. D. Sauzin, Résurgence paramétrique et exponentielle petitesse de lé́cart des séparatrices du pendule rapidement forcé, Ann. Ins. Fourier, 45 (1995), 453-511.
32. D. Sauzin, A new method for measuring the splitting of invariant manifolds, Ann. Sci. ńEcole Norm. Sup., 34 (2001), 159-221. (Pubitemid 33631318)
33. S. Smale, dieffomorphisms with many periodic points, in "dieffrential and Combinatorial Topology (A Symposium in Honor of Marston Morse)" 63-80. Princeton Univ. Press, Princeton, N.J., (1965).
34. J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings, in "Asymptotics Beyond All Orders (La Jolla, CA, 1991)" 284 of NATO Adv. Sci. Inst. Ser. B Phys., 187-195. Plenum, New York, (1991).
35. C. Simńo and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245.
36. D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russ. J. Math. Phys., 5 (1997), 63-98.