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Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom
1. | School of Mathematics, Institute for Advanced Study, Einstein Drive, Simonyi Hall, Princeton, New Jersey, 08540, United States |
References:
[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.
doi: 10.1088/0951-7715/19/6/011. |
[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. Differential Equations, 210 (2005), 106-134.
doi: 10.1016/j.jde.2004.10.017. |
[6] |
I. Baldomá, E. Fontich, M. Guàrdia and T. M. Seara, Exponentially small splitting of separatrices beyond melnikov analysis: Rigorous results, preprint, arXiv:1201.5152, 2011.
doi: 10.1016/j.jde.2012.09.003. |
[7] |
L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144 pp. |
[8] |
A. Delshams, V. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1997), 35-71.
doi: 10.1007/s002200050190. |
[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.
doi: 10.3934/dcds.2004.11.785. |
[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.
doi: 10.1016/0362-546X(93)90031-M. |
[13] |
E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations, 119 (1995), 310-335.
doi: 10.1006/jdeq.1995.1093. |
[14] |
V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, in "Seminar on Dynamical Systems (St. Petersburg, 1991)" 12 of Progr. Nonlinear Differential 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.
doi: 10.1016/S0167-2789(96)00133-9. |
[16] |
V. G. Gelfreich, Reference systems for splittings of separatrices, Nonlinearity, 10 (1997), 175-193.
doi: 10.1088/0951-7715/10/1/012. |
[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.
doi: 10.1007/s002200050579. |
[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.
doi: 10.1007/s00332-010-9068-8. |
[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.
doi: 10.1088/0951-7715/25/5/1367. |
[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).
doi: 10.1090/conm/081/986267. |
[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. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., 48 (1984), 197-204.
doi: 10.1016/0021-8928(84)90078-9. |
[28] |
C. Olivé, "Càlcul de L'escissió de Separatrius Usant Tècniques de Matching Complex I Ressurgència Aplicades a L'equació de Hamilton-Jacobi," Ph.D thesis, Universitat Politècnica 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 problème des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-270. |
[31] |
D. Sauzin, Résurgence paramétrique et exponentielle petitesse de l'écart 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. École Norm. Sup., 34 (2001), 159-221.
doi: 10.1016/S0012-9593(00)01063-6. |
[33] |
S. Smale, Diffeomorphisms with many periodic points, in "Differential 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. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245.
doi: 10.1088/0951-7715/22/5/012. |
[36] |
D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russ. J. Math. Phys., 5 (1997), 63-98. |
show all references
References:
[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.
doi: 10.1088/0951-7715/19/6/011. |
[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. Differential Equations, 210 (2005), 106-134.
doi: 10.1016/j.jde.2004.10.017. |
[6] |
I. Baldomá, E. Fontich, M. Guàrdia and T. M. Seara, Exponentially small splitting of separatrices beyond melnikov analysis: Rigorous results, preprint, arXiv:1201.5152, 2011.
doi: 10.1016/j.jde.2012.09.003. |
[7] |
L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144 pp. |
[8] |
A. Delshams, V. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1997), 35-71.
doi: 10.1007/s002200050190. |
[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.
doi: 10.3934/dcds.2004.11.785. |
[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.
doi: 10.1016/0362-546X(93)90031-M. |
[13] |
E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations, 119 (1995), 310-335.
doi: 10.1006/jdeq.1995.1093. |
[14] |
V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, in "Seminar on Dynamical Systems (St. Petersburg, 1991)" 12 of Progr. Nonlinear Differential 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.
doi: 10.1016/S0167-2789(96)00133-9. |
[16] |
V. G. Gelfreich, Reference systems for splittings of separatrices, Nonlinearity, 10 (1997), 175-193.
doi: 10.1088/0951-7715/10/1/012. |
[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.
doi: 10.1007/s002200050579. |
[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.
doi: 10.1007/s00332-010-9068-8. |
[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.
doi: 10.1088/0951-7715/25/5/1367. |
[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).
doi: 10.1090/conm/081/986267. |
[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. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., 48 (1984), 197-204.
doi: 10.1016/0021-8928(84)90078-9. |
[28] |
C. Olivé, "Càlcul de L'escissió de Separatrius Usant Tècniques de Matching Complex I Ressurgència Aplicades a L'equació de Hamilton-Jacobi," Ph.D thesis, Universitat Politècnica 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 problème des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-270. |
[31] |
D. Sauzin, Résurgence paramétrique et exponentielle petitesse de l'écart 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. École Norm. Sup., 34 (2001), 159-221.
doi: 10.1016/S0012-9593(00)01063-6. |
[33] |
S. Smale, Diffeomorphisms with many periodic points, in "Differential 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. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245.
doi: 10.1088/0951-7715/22/5/012. |
[36] |
D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russ. J. Math. Phys., 5 (1997), 63-98. |
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