# American Institute of Mathematical Sciences

July  2013, 33(7): 2829-2859. doi: 10.3934/dcds.2013.33.2829

## 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

Received  March 2012 Revised  April 2012 Published  January 2013

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.
Citation: Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
##### 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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##### 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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