# American Institute of Mathematical Sciences

2014, 21: 41-61. doi: 10.3934/era.2014.21.41

## Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

 1 Departament de Matemática Aplicada I, ETSEIB-UPC, 08028 Barcelona 2 Institut für Mathematik, MA 7-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany 3 Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028

Received  May 2013 Revised  November 2013 Published  May 2014

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic irrational number, or a 3-dimensional torus with a frequency vector $\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number. Applying the Poincaré--Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.
Citation: Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez. Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies. Electronic Research Announcements, 2014, 21: 41-61. doi: 10.3934/era.2014.21.41
##### References:
 [1] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5 (1964), 581-585. Google Scholar [2] 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.  Google Scholar [3] G. Benettin, A. Carati and F. Fassò, On the conservation of adiabatic invariants for a system of coupled rotators, Phys. D, 104 (1997), 253-268. doi: 10.1016/S0167-2789(97)00295-9.  Google Scholar [4] G. Benettin, A. Carati and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants, Nonlinearity, 10 (1997), 479-505. doi: 10.1088/0951-7715/10/2/011.  Google Scholar [5] I. Baldomá, E. Fontich, M. Guardia and T. M. Seara, Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results, J. Differential Equations, 253 (2012), 3304-3439. doi: 10.1016/j.jde.2012.09.003.  Google Scholar [6] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957.  Google Scholar [7] C. Chandre, Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 457-465. doi: 10.3934/dcdsb.2002.2.457.  Google Scholar [8] A. Delshams and P. Gutiérrez, Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 433-476. doi: 10.1007/s003329910016.  Google Scholar [9] A. Delshams and P. Gutiérrez, Homoclinic orbits to invariant tori in Hamiltonian systems, in Multiple-Time-Scale Dynamical Systems (Minneapolis, MN, 1997) (eds. C. K. R. T. Jones and A. I. Khibnik), IMA Vol. Math. Appl., 122, Springer, New York, 2001, 1-27. doi: 10.1007/978-1-4613-0117-2_1.  Google Scholar [10] A. Delshams and P. Gutiérrez, Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 87-121; translation in J. Math. Sci. (N.Y.), 128 (2005), 2726-2746. doi: 10.1007/s10958-005-0224-x.  Google Scholar [11] A. Delshams and P. Gutiérrez, Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits, Discrete Contin. Dyn. Syst., 11 (2004), 757-783. doi: 10.3934/dcds.2004.11.757.  Google Scholar [12] A. Delshams, V. G. 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.  Google Scholar [13] A. Delshams, P. Gutiérrez and T. M. Seara, Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds, Discrete Contin. Dyn. Syst., 11 (2004), 785-826. doi: 10.3934/dcds.2004.11.785.  Google Scholar [14] A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8 (1998), 317-352. doi: 10.1007/s003329900054.  Google Scholar [15] 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. doi: 10.1007/BF02096956.  Google Scholar [16] 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), paper 4, 40 pp.  Google Scholar [17] L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. (N. S.), 25 (1994), 57-76. doi: 10.1007/BF01232935.  Google Scholar [18] 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.  Google Scholar [19] E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations, 119 (1995), 310-335. doi: 10.1006/jdeq.1995.1093.  Google Scholar [20] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563.  Google Scholar [21] G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys., 6 (1994), 343-411. doi: 10.1142/S0129055X9400016X.  Google Scholar [22] 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.  Google Scholar [23] G. Gallavotti, G. Gentile and V. Mastropietro, Mel'nikov's approximation dominance. Some examples, Rev. Math. Phys., 11 (1999), 451-461. doi: 10.1142/S0129055X99000167.  Google Scholar [24] M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dyn. Syst., 14 (2006), 295-328.  Google Scholar [25] M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori, J. Differential Equations, 193 (2003), 49-74. doi: 10.1016/S0022-0396(03)00065-2.  Google Scholar [26] 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.  Google Scholar [27] D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms, Ergodic Theory Dynam. Systems, 20 (2000), 1711-1733. doi: 10.1017/S014338570000095X.  Google Scholar [28] D. M. Hardcastle and K. Khanin, Continued fractions and the $d$-dimensional Gauss transformation, Comm. Math. Phys., 215 (2001), 487-515. doi: 10.1007/s002200000290.  Google Scholar [29] P. Holmes, J. E. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, in Hamiltonian Dynamical Systems (Boulder, CO, 1987), Contemp. Math., 81, Amer. Math. Soc., Providence, RI, 1988, 213-244. doi: 10.1090/conm/081/986267.  Google Scholar [30] K. Khanin, J. Lopes Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows, Nonlinearity, 19 (2006), 2727-2753. doi: 10.1088/0951-7715/19/12/001.  Google Scholar [31] K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231. doi: 10.1007/s00220-006-0125-y.  Google Scholar [32] H. Koch, A renormalization group for Hamiltonians, with applications to KAM theory, Ergodic Theory Dynam. Systems, 19 (1999), 475-521. doi: 10.1017/S0143385799130128.  Google Scholar [33] D. V. Kosygin, Multidimensional KAM theory from the renormalization group viewpoint, in Dynamical Systems and Statistical Mechanics (Moscow, 1991), Translated from the Russian by V. E. Nazaikinskiĭ, Adv. Soviet Math., 3, Amer. Math. Soc., Providence, RI, 1991, 99-129.  Google Scholar [34] V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 25-55, 285. doi: 10.1007/s10958-005-0219-7.  Google Scholar [35] P. Lochak and C. Meunier, Multiphase averaging for classical systems, with applications to adiabatic theorems, Appl. Math. Sci., 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.  Google Scholar [36] 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). doi: 10.1090/memo/0775.  Google Scholar [37] P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar [38] E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems, Lect. Notes in Math., 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar [39] J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector, Nonlinearity, 15 (2002), 647-664. doi: 10.1088/0951-7715/15/3/307.  Google Scholar [40] V. K. Mel'nikov, On the stability of a center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57.  Google Scholar [41] P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 165-207. doi: 10.3934/dcds.2011.31.165.  Google Scholar [42] P. Martín, D. Sauzin and T. M. Seara, Exponentially small splitting of separatrices in the perturbed McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 301-372. doi: 10.3934/dcds.2011.31.301.  Google Scholar [43] A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139; translated from Prikl. Mat. Mekh., 48 (1984), 197-204. doi: 10.1016/0021-8928(84)90078-9.  Google Scholar [44] L. Niederman, Dynamics around simple resonant tori in nearly integrable Hamiltonian systems, J. Differential Equations, 161 (2000), 1-41. doi: 10.1006/jdeq.1999.3692.  Google Scholar [45] C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation, Ann. Inst. Fourier (Grenoble), 53 (2003), 1185-1235. doi: 10.5802/aif.1977.  Google Scholar [46] H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. Google Scholar [47] A. Pronin and D. V. Treschev, Continuous averaging in multi-frequency slow-fast systems, Regul. Chaotic Dyn., 5 (2000), 157-170. doi: 10.1070/rd2000v005n02ABEH000138.  Google Scholar [48] M. Rudnev and S. Wiggins, On a homoclinic splitting problem, Regul. Chaotic Dyn., 5 (2000), 227-242. doi: 10.1070/rd2000v005n02ABEH000146.  Google Scholar [49] D. Sauzin, A new method for measuring the splitting of invariant manifolds, Ann. Sci. École Norm. Sup. (4), 34 (2001), 159-221. doi: 10.1016/S0012-9593(00)01063-6.  Google Scholar [50] C. Simó, Averaging under fast quasiperiodic forcing, in Hamiltonian Mechanics (ed. J. Seimenis) (Toruń, 1993), NATO ASI Ser. B: Phys., 331, Plenum, New York, 1994, 13-34.  Google Scholar [51] C. Simó and C. Valls, A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters, Nonlinearity, 14 (2001), 1707-1760. doi: 10.1088/0951-7715/14/6/316.  Google Scholar [52] D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98.  Google Scholar

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##### References:
 [1] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5 (1964), 581-585. Google Scholar [2] 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.  Google Scholar [3] G. Benettin, A. Carati and F. Fassò, On the conservation of adiabatic invariants for a system of coupled rotators, Phys. D, 104 (1997), 253-268. doi: 10.1016/S0167-2789(97)00295-9.  Google Scholar [4] G. Benettin, A. Carati and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants, Nonlinearity, 10 (1997), 479-505. doi: 10.1088/0951-7715/10/2/011.  Google Scholar [5] I. Baldomá, E. Fontich, M. Guardia and T. M. Seara, Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results, J. Differential Equations, 253 (2012), 3304-3439. doi: 10.1016/j.jde.2012.09.003.  Google Scholar [6] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957.  Google Scholar [7] C. Chandre, Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 457-465. doi: 10.3934/dcdsb.2002.2.457.  Google Scholar [8] A. Delshams and P. Gutiérrez, Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 433-476. doi: 10.1007/s003329910016.  Google Scholar [9] A. Delshams and P. Gutiérrez, Homoclinic orbits to invariant tori in Hamiltonian systems, in Multiple-Time-Scale Dynamical Systems (Minneapolis, MN, 1997) (eds. C. K. R. T. Jones and A. I. Khibnik), IMA Vol. Math. Appl., 122, Springer, New York, 2001, 1-27. doi: 10.1007/978-1-4613-0117-2_1.  Google Scholar [10] A. Delshams and P. Gutiérrez, Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 87-121; translation in J. Math. Sci. (N.Y.), 128 (2005), 2726-2746. doi: 10.1007/s10958-005-0224-x.  Google Scholar [11] A. Delshams and P. Gutiérrez, Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits, Discrete Contin. Dyn. Syst., 11 (2004), 757-783. doi: 10.3934/dcds.2004.11.757.  Google Scholar [12] A. Delshams, V. G. 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.  Google Scholar [13] A. Delshams, P. Gutiérrez and T. M. Seara, Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds, Discrete Contin. Dyn. Syst., 11 (2004), 785-826. doi: 10.3934/dcds.2004.11.785.  Google Scholar [14] A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8 (1998), 317-352. doi: 10.1007/s003329900054.  Google Scholar [15] 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. doi: 10.1007/BF02096956.  Google Scholar [16] 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), paper 4, 40 pp.  Google Scholar [17] L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. (N. S.), 25 (1994), 57-76. doi: 10.1007/BF01232935.  Google Scholar [18] 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.  Google Scholar [19] E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations, 119 (1995), 310-335. doi: 10.1006/jdeq.1995.1093.  Google Scholar [20] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563.  Google Scholar [21] G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys., 6 (1994), 343-411. doi: 10.1142/S0129055X9400016X.  Google Scholar [22] 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.  Google Scholar [23] G. Gallavotti, G. Gentile and V. Mastropietro, Mel'nikov's approximation dominance. Some examples, Rev. Math. Phys., 11 (1999), 451-461. doi: 10.1142/S0129055X99000167.  Google Scholar [24] M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dyn. Syst., 14 (2006), 295-328.  Google Scholar [25] M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori, J. Differential Equations, 193 (2003), 49-74. doi: 10.1016/S0022-0396(03)00065-2.  Google Scholar [26] 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.  Google Scholar [27] D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms, Ergodic Theory Dynam. Systems, 20 (2000), 1711-1733. doi: 10.1017/S014338570000095X.  Google Scholar [28] D. M. Hardcastle and K. Khanin, Continued fractions and the $d$-dimensional Gauss transformation, Comm. Math. Phys., 215 (2001), 487-515. doi: 10.1007/s002200000290.  Google Scholar [29] P. Holmes, J. E. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, in Hamiltonian Dynamical Systems (Boulder, CO, 1987), Contemp. Math., 81, Amer. Math. Soc., Providence, RI, 1988, 213-244. doi: 10.1090/conm/081/986267.  Google Scholar [30] K. Khanin, J. Lopes Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows, Nonlinearity, 19 (2006), 2727-2753. doi: 10.1088/0951-7715/19/12/001.  Google Scholar [31] K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231. doi: 10.1007/s00220-006-0125-y.  Google Scholar [32] H. Koch, A renormalization group for Hamiltonians, with applications to KAM theory, Ergodic Theory Dynam. Systems, 19 (1999), 475-521. doi: 10.1017/S0143385799130128.  Google Scholar [33] D. V. Kosygin, Multidimensional KAM theory from the renormalization group viewpoint, in Dynamical Systems and Statistical Mechanics (Moscow, 1991), Translated from the Russian by V. E. Nazaikinskiĭ, Adv. Soviet Math., 3, Amer. Math. Soc., Providence, RI, 1991, 99-129.  Google Scholar [34] V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 25-55, 285. doi: 10.1007/s10958-005-0219-7.  Google Scholar [35] P. Lochak and C. Meunier, Multiphase averaging for classical systems, with applications to adiabatic theorems, Appl. Math. Sci., 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.  Google Scholar [36] 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). doi: 10.1090/memo/0775.  Google Scholar [37] P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar [38] E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems, Lect. Notes in Math., 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar [39] J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector, Nonlinearity, 15 (2002), 647-664. doi: 10.1088/0951-7715/15/3/307.  Google Scholar [40] V. K. Mel'nikov, On the stability of a center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57.  Google Scholar [41] P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 165-207. doi: 10.3934/dcds.2011.31.165.  Google Scholar [42] P. Martín, D. Sauzin and T. M. Seara, Exponentially small splitting of separatrices in the perturbed McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 301-372. doi: 10.3934/dcds.2011.31.301.  Google Scholar [43] A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139; translated from Prikl. Mat. Mekh., 48 (1984), 197-204. doi: 10.1016/0021-8928(84)90078-9.  Google Scholar [44] L. Niederman, Dynamics around simple resonant tori in nearly integrable Hamiltonian systems, J. Differential Equations, 161 (2000), 1-41. doi: 10.1006/jdeq.1999.3692.  Google Scholar [45] C. Olivé, D. 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