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

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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

Abstract
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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.

Keywords: complex matching., Nearly integrable Hamiltonian systems, exponentially small splitting of separatrices, Melnikov method.

Mathematics Subject Classification: 37J40, 37J4.

Citation: Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems - A, 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., 3 (1988). Google Scholar
[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. Google Scholar
[3]	I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems,, Nonlinearity, 19 (2006), 1415. doi: 10.1088/0951-7715/19/6/011. Google Scholar
[4]	I. Baldomá and E. Fontich, Exponentially small splitting of invariant manifolds of parabolic points,, Mem. Amer. Math. Soc., 167 (2004). Google Scholar
[5]	I. Baldomá and E. Fontich, Exponentially small splitting of separatrices in a weakly hyperbolic case,, J. Differential Equations, 210 (2005), 106. doi: 10.1016/j.jde.2004.10.017. Google Scholar
[6]	I. Baldomá, E. Fontich, M. Guàrdia and T. M. Seara, Exponentially small splitting of separatrices beyond melnikov analysis: Rigorous results,, preprint, (2011). doi: 10.1016/j.jde.2012.09.003. Google Scholar
[7]	L. Chierchia and G. Gallavotti, Drift and diffusion in phase space,, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994). Google Scholar
[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. doi: 10.1007/s002200050190. Google Scholar
[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. doi: 10.3934/dcds.2004.11.785. Google Scholar
[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. Google Scholar
[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). Google Scholar
[12]	E. Fontich, Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations,, Nonlinear Anal., 20 (1993), 733. doi: 10.1016/0362-546X(93)90031-M. Google Scholar
[13]	E. Fontich, Rapidly forced planar vector fields and splitting of separatrices,, J. Differential Equations, 119 (1995), 310. doi: 10.1006/jdeq.1995.1093. Google Scholar
[14]	V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum,, in, 12 (1994), 47. Google Scholar
[15]	V. G. Gelfreich, Melnikov method and exponentially small splitting of separatrices,, Phys. D, 101 (1997), 227. doi: 10.1016/S0167-2789(96)00133-9. Google Scholar
[16]	V. G. Gelfreich, Reference systems for splittings of separatrices,, Nonlinearity, 10 (1997), 175. doi: 10.1088/0951-7715/10/1/012. Google Scholar
[17]	V. G. Gelfreich, Separatrix splitting for a high-frequency perturbation of the pendulum,, Russ. J. Math. Phys., 7 (2000), 48. Google Scholar
[18]	G. Gallavotti, G. Gentile and V. Mastropietro, Separatrix splitting for systems with three time scales,, Comm. Math. Phys., 202 (1999), 197. doi: 10.1007/s002200050579. Google Scholar
[19]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983). Google Scholar
[20]	M. Guardia, C. Olivé and T. Seara, Exponentially small splitting for the pendulum: A classical problem revisited,, J. Nonlinear Sci., 20 (2010), 595. doi: 10.1007/s00332-010-9068-8. Google Scholar
[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. doi: 10.1088/0951-7715/25/5/1367. Google Scholar
[22]	P. Holmes, J. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations,, in, 81 (1988). doi: 10.1090/conm/081/986267. Google Scholar
[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. Google Scholar
[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). Google Scholar
[25]	V. K. Melnikov, On the stability of the center for time periodic perturbations,, Trans. Moscow Math. Soc., 12 (1963), 1. Google Scholar
[26]	J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1962), 1. Google Scholar
[27]	A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase,, Prikl. Mat. Mekh., 48 (1984), 197. doi: 10.1016/0021-8928(84)90078-9. Google Scholar
[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, (2006). Google Scholar
[29]	C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, in, 53 (2003), 1185. Google Scholar
[30]	H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar
[31]	D. Sauzin, Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé,, Ann. Ins. Fourier, 45 (1995), 453. Google Scholar
[32]	D. Sauzin, A new method for measuring the splitting of invariant manifolds,, Ann. Sci. École Norm. Sup., 34 (2001), 159. doi: 10.1016/S0012-9593(00)01063-6. Google Scholar
[33]	S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar
[34]	J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings,, in, 284 (1991), 187. Google Scholar
[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. doi: 10.1088/0951-7715/22/5/012. Google Scholar
[36]	D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point,, Russ. J. Math. Phys., 5 (1997), 63. Google Scholar

show all references

References:

[1]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Dynamical Systems III,", 3 of Encyclopaedia Math. Sci., 3 (1988). Google Scholar
[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. Google Scholar
[3]	I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems,, Nonlinearity, 19 (2006), 1415. doi: 10.1088/0951-7715/19/6/011. Google Scholar
[4]	I. Baldomá and E. Fontich, Exponentially small splitting of invariant manifolds of parabolic points,, Mem. Amer. Math. Soc., 167 (2004). Google Scholar
[5]	I. Baldomá and E. Fontich, Exponentially small splitting of separatrices in a weakly hyperbolic case,, J. Differential Equations, 210 (2005), 106. doi: 10.1016/j.jde.2004.10.017. Google Scholar
[6]	I. Baldomá, E. Fontich, M. Guàrdia and T. M. Seara, Exponentially small splitting of separatrices beyond melnikov analysis: Rigorous results,, preprint, (2011). doi: 10.1016/j.jde.2012.09.003. Google Scholar
[7]	L. Chierchia and G. Gallavotti, Drift and diffusion in phase space,, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994). Google Scholar
[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. doi: 10.1007/s002200050190. Google Scholar
[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. doi: 10.3934/dcds.2004.11.785. Google Scholar
[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. Google Scholar
[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). Google Scholar
[12]	E. Fontich, Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations,, Nonlinear Anal., 20 (1993), 733. doi: 10.1016/0362-546X(93)90031-M. Google Scholar
[13]	E. Fontich, Rapidly forced planar vector fields and splitting of separatrices,, J. Differential Equations, 119 (1995), 310. doi: 10.1006/jdeq.1995.1093. Google Scholar
[14]	V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum,, in, 12 (1994), 47. Google Scholar
[15]	V. G. Gelfreich, Melnikov method and exponentially small splitting of separatrices,, Phys. D, 101 (1997), 227. doi: 10.1016/S0167-2789(96)00133-9. Google Scholar
[16]	V. G. Gelfreich, Reference systems for splittings of separatrices,, Nonlinearity, 10 (1997), 175. doi: 10.1088/0951-7715/10/1/012. Google Scholar
[17]	V. G. Gelfreich, Separatrix splitting for a high-frequency perturbation of the pendulum,, Russ. J. Math. Phys., 7 (2000), 48. Google Scholar
[18]	G. Gallavotti, G. Gentile and V. Mastropietro, Separatrix splitting for systems with three time scales,, Comm. Math. Phys., 202 (1999), 197. doi: 10.1007/s002200050579. Google Scholar
[19]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983). Google Scholar
[20]	M. Guardia, C. Olivé and T. Seara, Exponentially small splitting for the pendulum: A classical problem revisited,, J. Nonlinear Sci., 20 (2010), 595. doi: 10.1007/s00332-010-9068-8. Google Scholar
[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. doi: 10.1088/0951-7715/25/5/1367. Google Scholar
[22]	P. Holmes, J. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations,, in, 81 (1988). doi: 10.1090/conm/081/986267. Google Scholar
[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. Google Scholar
[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). Google Scholar
[25]	V. K. Melnikov, On the stability of the center for time periodic perturbations,, Trans. Moscow Math. Soc., 12 (1963), 1. Google Scholar
[26]	J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1962), 1. Google Scholar
[27]	A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase,, Prikl. Mat. Mekh., 48 (1984), 197. doi: 10.1016/0021-8928(84)90078-9. Google Scholar
[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, (2006). Google Scholar
[29]	C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, in, 53 (2003), 1185. Google Scholar
[30]	H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar
[31]	D. Sauzin, Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé,, Ann. Ins. Fourier, 45 (1995), 453. Google Scholar
[32]	D. Sauzin, A new method for measuring the splitting of invariant manifolds,, Ann. Sci. École Norm. Sup., 34 (2001), 159. doi: 10.1016/S0012-9593(00)01063-6. Google Scholar
[33]	S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar
[34]	J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings,, in, 284 (1991), 187. Google Scholar
[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. doi: 10.1088/0951-7715/22/5/012. Google Scholar
[36]	D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point,, Russ. J. Math. Phys., 5 (1997), 63. Google Scholar

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