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 für Mathematik, MA 7-2, Technische Universität 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,, \emph{Soviet Math. Dokl.}, 5 (1964), 581.

[2]

I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems,, \emph{Nonlinearity}, 19 (2006), 1415. doi: 10.1088/0951-7715/19/6/011.

[3]

G. Benettin, A. Carati and F. Fassò, On the conservation of adiabatic invariants for a system of coupled rotators,, \emph{Phys. D}, 104 (1997), 253. doi: 10.1016/S0167-2789(97)00295-9.

[4]

G. Benettin, A. Carati and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants,, \emph{Nonlinearity}, 10 (1997), 479. doi: 10.1088/0951-7715/10/2/011.

[5]

I. Baldomá, E. Fontich, M. Guardia and T. M. Seara, Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results,, \emph{J. Differential Equations}, 253 (2012), 3304. doi: 10.1016/j.jde.2012.09.003.

[6]

J. W. S. Cassels, An Introduction to Diophantine Approximation,, Cambridge Univ. Press, (1957).

[7]

C. Chandre, Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 2 (2002), 457. doi: 10.3934/dcdsb.2002.2.457.

[8]

A. Delshams and P. Gutiérrez, Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems,, \emph{J. Nonlinear Sci.}, 10 (2000), 433. doi: 10.1007/s003329910016.

[9]

A. Delshams and P. Gutiérrez, Homoclinic orbits to invariant tori in Hamiltonian systems,, in \emph{Multiple-Time-Scale Dynamical Systems} (Minneapolis, (1997), 1. doi: 10.1007/978-1-4613-0117-2_1.

[10]

A. Delshams and P. Gutiérrez, Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 300 (2003), 87. doi: 10.1007/s10958-005-0224-x.

[11]

A. Delshams and P. Gutiérrez, Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits,, \emph{Discrete Contin. Dyn. Syst.}, 11 (2004), 757. doi: 10.3934/dcds.2004.11.757.

[12]

A. Delshams, V. G. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing,, \emph{Comm. Math. Phys.}, 189 (1997), 35. doi: 10.1007/s002200050190.

[13]

A. Delshams, P. Gutiérrez and T. M. Seara, Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds,, \emph{Discrete Contin. Dyn. Syst.}, 11 (2004), 785. doi: 10.3934/dcds.2004.11.785.

[14]

A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps,, \emph{J. Nonlinear Sci.}, 8 (1998), 317. doi: 10.1007/s003329900054.

[15]

A. Delshams and T. M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum,, \emph{Comm. Math. Phys.}, 150 (1992), 433. doi: 10.1007/BF02096956.

[16]

A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom,, \emph{Math. Phys. Electron. J.}, 3 (1997).

[17]

L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems,, \emph{Bol. Soc. Brasil. Mat. (N. S.)}, 25 (1994), 57. doi: 10.1007/BF01232935.

[18]

E. Fontich, Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations,, \emph{Nonlinear Anal.}, 20 (1993), 733. doi: 10.1016/0362-546X(93)90031-M.

[19]

E. Fontich, Rapidly forced planar vector fields and splitting of separatrices,, \emph{J. Differential Equations}, 119 (1995), 310. doi: 10.1006/jdeq.1995.1093.

[20]

E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms,, \emph{Ergodic Theory Dynam. Systems}, 10 (1990), 295. doi: 10.1017/S0143385700005563.

[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,, \emph{Rev. Math. Phys.}, 6 (1994), 343. doi: 10.1142/S0129055X9400016X.

[22]

V. G. Gelfreich, Melnikov method and exponentially small splitting of separatrices,, \emph{Phys. D}, 101 (1997), 227. doi: 10.1016/S0167-2789(96)00133-9.

[23]

G. Gallavotti, G. Gentile and V. Mastropietro, Mel'nikov's approximation dominance. Some examples,, \emph{Rev. Math. Phys.}, 11 (1999), 451. doi: 10.1142/S0129055X99000167.

[24]

M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems,, \emph{Discrete Contin. Dyn. Syst.}, 14 (2006), 295.

[25]

M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori,, \emph{J. Differential Equations}, 193 (2003), 49. doi: 10.1016/S0022-0396(03)00065-2.

[26]

M. Guardia and T. M. Seara, Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation,, \emph{Nonlinearity}, 25 (2012), 1367. doi: 10.1088/0951-7715/25/5/1367.

[27]

D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms,, \emph{Ergodic Theory Dynam. Systems}, 20 (2000), 1711. doi: 10.1017/S014338570000095X.

[28]

D. M. Hardcastle and K. Khanin, Continued fractions and the $d$-dimensional Gauss transformation,, \emph{Comm. Math. Phys.}, 215 (2001), 487. doi: 10.1007/s002200000290.

[29]

P. Holmes, J. E. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations,, in \emph{Hamiltonian Dynamical Systems} (Boulder, (1987), 213. doi: 10.1090/conm/081/986267.

[30]

K. Khanin, J. Lopes Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows,, \emph{Nonlinearity}, 19 (2006), 2727. doi: 10.1088/0951-7715/19/12/001.

[31]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory,, \emph{Comm. Math. Phys.}, 270 (2007), 197. doi: 10.1007/s00220-006-0125-y.

[32]

H. Koch, A renormalization group for Hamiltonians, with applications to KAM theory,, \emph{Ergodic Theory Dynam. Systems}, 19 (1999), 475. doi: 10.1017/S0143385799130128.

[33]

D. V. Kosygin, Multidimensional KAM theory from the renormalization group viewpoint,, in \emph{Dynamical Systems and Statistical Mechanics} (Moscow, (1991), 99.

[34]

V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 300 (2003), 25. doi: 10.1007/s10958-005-0219-7.

[35]

P. Lochak and C. Meunier, Multiphase averaging for classical systems, with applications to adiabatic theorems,, Appl. Math. Sci., (1988). doi: 10.1007/978-1-4612-1044-3.

[36]

P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems,, \emph{Mem. Amer. Math. Soc.}, 163 (2003). doi: 10.1090/memo/0775.

[37]

P. Lochak, Canonical perturbation theory via simultaneous approximation,, \emph{Russian Math. Surveys}, 47 (1992), 57. doi: 10.1070/RM1992v047n06ABEH000965.

[38]

E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems,, Lect. Notes in Math., (1741). doi: 10.1007/BFb0104102.

[39]

J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector,, \emph{Nonlinearity}, 15 (2002), 647. doi: 10.1088/0951-7715/15/3/307.

[40]

V. K. Mel'nikov, On the stability of a center for time-periodic perturbations,, \emph{Trans. Moscow Math. Soc.}, 12 (1963), 1.

[41]

P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 165. doi: 10.3934/dcds.2011.31.165.

[42]

P. Martín, D. Sauzin and T. M. Seara, Exponentially small splitting of separatrices in the perturbed McMillan map,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 301. doi: 10.3934/dcds.2011.31.301.

[43]

A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase,, \emph{J. Appl. Math. Mech.}, 48 (1984), 133. doi: 10.1016/0021-8928(84)90078-9.

[44]

L. Niederman, Dynamics around simple resonant tori in nearly integrable Hamiltonian systems,, \emph{J. Differential Equations}, 161 (2000), 1. doi: 10.1006/jdeq.1999.3692.

[45]

C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, \emph{Ann. Inst. Fourier (Grenoble)}, 53 (2003), 1185. doi: 10.5802/aif.1977.

[46]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, \emph{Acta Math.}, 13 (1890), 1.

[47]

A. Pronin and D. V. Treschev, Continuous averaging in multi-frequency slow-fast systems,, \emph{Regul. Chaotic Dyn.}, 5 (2000), 157. doi: 10.1070/rd2000v005n02ABEH000138.

[48]

M. Rudnev and S. Wiggins, On a homoclinic splitting problem,, \emph{Regul. Chaotic Dyn.}, 5 (2000), 227. doi: 10.1070/rd2000v005n02ABEH000146.

[49]

D. Sauzin, A new method for measuring the splitting of invariant manifolds,, \emph{Ann. Sci. École Norm. Sup. (4)}, 34 (2001), 159. doi: 10.1016/S0012-9593(00)01063-6.

[50]

C. Simó, Averaging under fast quasiperiodic forcing,, in \emph{Hamiltonian Mechanics} (ed. J. Seimenis) (Toruń, (1993), 13.

[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,, \emph{Nonlinearity}, 14 (2001), 1707. doi: 10.1088/0951-7715/14/6/316.

[52]

D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point,, \emph{Russian J. Math. Phys.}, 5 (1997), 63.

show all references

References:
[1]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom,, \emph{Soviet Math. Dokl.}, 5 (1964), 581.

[2]

I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems,, \emph{Nonlinearity}, 19 (2006), 1415. doi: 10.1088/0951-7715/19/6/011.

[3]

G. Benettin, A. Carati and F. Fassò, On the conservation of adiabatic invariants for a system of coupled rotators,, \emph{Phys. D}, 104 (1997), 253. doi: 10.1016/S0167-2789(97)00295-9.

[4]

G. Benettin, A. Carati and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants,, \emph{Nonlinearity}, 10 (1997), 479. doi: 10.1088/0951-7715/10/2/011.

[5]

I. Baldomá, E. Fontich, M. Guardia and T. M. Seara, Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results,, \emph{J. Differential Equations}, 253 (2012), 3304. doi: 10.1016/j.jde.2012.09.003.

[6]

J. W. S. Cassels, An Introduction to Diophantine Approximation,, Cambridge Univ. Press, (1957).

[7]

C. Chandre, Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 2 (2002), 457. doi: 10.3934/dcdsb.2002.2.457.

[8]

A. Delshams and P. Gutiérrez, Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems,, \emph{J. Nonlinear Sci.}, 10 (2000), 433. doi: 10.1007/s003329910016.

[9]

A. Delshams and P. Gutiérrez, Homoclinic orbits to invariant tori in Hamiltonian systems,, in \emph{Multiple-Time-Scale Dynamical Systems} (Minneapolis, (1997), 1. doi: 10.1007/978-1-4613-0117-2_1.

[10]

A. Delshams and P. Gutiérrez, Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 300 (2003), 87. doi: 10.1007/s10958-005-0224-x.

[11]

A. Delshams and P. Gutiérrez, Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits,, \emph{Discrete Contin. Dyn. Syst.}, 11 (2004), 757. doi: 10.3934/dcds.2004.11.757.

[12]

A. Delshams, V. G. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing,, \emph{Comm. Math. Phys.}, 189 (1997), 35. doi: 10.1007/s002200050190.

[13]

A. Delshams, P. Gutiérrez and T. M. Seara, Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds,, \emph{Discrete Contin. Dyn. Syst.}, 11 (2004), 785. doi: 10.3934/dcds.2004.11.785.

[14]

A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps,, \emph{J. Nonlinear Sci.}, 8 (1998), 317. doi: 10.1007/s003329900054.

[15]

A. Delshams and T. M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum,, \emph{Comm. Math. Phys.}, 150 (1992), 433. doi: 10.1007/BF02096956.

[16]

A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom,, \emph{Math. Phys. Electron. J.}, 3 (1997).

[17]

L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems,, \emph{Bol. Soc. Brasil. Mat. (N. S.)}, 25 (1994), 57. doi: 10.1007/BF01232935.

[18]

E. Fontich, Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations,, \emph{Nonlinear Anal.}, 20 (1993), 733. doi: 10.1016/0362-546X(93)90031-M.

[19]

E. Fontich, Rapidly forced planar vector fields and splitting of separatrices,, \emph{J. Differential Equations}, 119 (1995), 310. doi: 10.1006/jdeq.1995.1093.

[20]

E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms,, \emph{Ergodic Theory Dynam. Systems}, 10 (1990), 295. doi: 10.1017/S0143385700005563.

[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,, \emph{Rev. Math. Phys.}, 6 (1994), 343. doi: 10.1142/S0129055X9400016X.

[22]

V. G. Gelfreich, Melnikov method and exponentially small splitting of separatrices,, \emph{Phys. D}, 101 (1997), 227. doi: 10.1016/S0167-2789(96)00133-9.

[23]

G. Gallavotti, G. Gentile and V. Mastropietro, Mel'nikov's approximation dominance. Some examples,, \emph{Rev. Math. Phys.}, 11 (1999), 451. doi: 10.1142/S0129055X99000167.

[24]

M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems,, \emph{Discrete Contin. Dyn. Syst.}, 14 (2006), 295.

[25]

M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori,, \emph{J. Differential Equations}, 193 (2003), 49. doi: 10.1016/S0022-0396(03)00065-2.

[26]

M. Guardia and T. M. Seara, Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation,, \emph{Nonlinearity}, 25 (2012), 1367. doi: 10.1088/0951-7715/25/5/1367.

[27]

D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms,, \emph{Ergodic Theory Dynam. Systems}, 20 (2000), 1711. doi: 10.1017/S014338570000095X.

[28]

D. M. Hardcastle and K. Khanin, Continued fractions and the $d$-dimensional Gauss transformation,, \emph{Comm. Math. Phys.}, 215 (2001), 487. doi: 10.1007/s002200000290.

[29]

P. Holmes, J. E. Marsden and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations,, in \emph{Hamiltonian Dynamical Systems} (Boulder, (1987), 213. doi: 10.1090/conm/081/986267.

[30]

K. Khanin, J. Lopes Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows,, \emph{Nonlinearity}, 19 (2006), 2727. doi: 10.1088/0951-7715/19/12/001.

[31]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory,, \emph{Comm. Math. Phys.}, 270 (2007), 197. doi: 10.1007/s00220-006-0125-y.

[32]

H. Koch, A renormalization group for Hamiltonians, with applications to KAM theory,, \emph{Ergodic Theory Dynam. Systems}, 19 (1999), 475. doi: 10.1017/S0143385799130128.

[33]

D. V. Kosygin, Multidimensional KAM theory from the renormalization group viewpoint,, in \emph{Dynamical Systems and Statistical Mechanics} (Moscow, (1991), 99.

[34]

V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 300 (2003), 25. doi: 10.1007/s10958-005-0219-7.

[35]

P. Lochak and C. Meunier, Multiphase averaging for classical systems, with applications to adiabatic theorems,, Appl. Math. Sci., (1988). doi: 10.1007/978-1-4612-1044-3.

[36]

P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems,, \emph{Mem. Amer. Math. Soc.}, 163 (2003). doi: 10.1090/memo/0775.

[37]

P. Lochak, Canonical perturbation theory via simultaneous approximation,, \emph{Russian Math. Surveys}, 47 (1992), 57. doi: 10.1070/RM1992v047n06ABEH000965.

[38]

E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems,, Lect. Notes in Math., (1741). doi: 10.1007/BFb0104102.

[39]

J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector,, \emph{Nonlinearity}, 15 (2002), 647. doi: 10.1088/0951-7715/15/3/307.

[40]

V. K. Mel'nikov, On the stability of a center for time-periodic perturbations,, \emph{Trans. Moscow Math. Soc.}, 12 (1963), 1.

[41]

P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 165. doi: 10.3934/dcds.2011.31.165.

[42]

P. Martín, D. Sauzin and T. M. Seara, Exponentially small splitting of separatrices in the perturbed McMillan map,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 301. doi: 10.3934/dcds.2011.31.301.

[43]

A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase,, \emph{J. Appl. Math. Mech.}, 48 (1984), 133. doi: 10.1016/0021-8928(84)90078-9.

[44]

L. Niederman, Dynamics around simple resonant tori in nearly integrable Hamiltonian systems,, \emph{J. Differential Equations}, 161 (2000), 1. doi: 10.1006/jdeq.1999.3692.

[45]

C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, \emph{Ann. Inst. Fourier (Grenoble)}, 53 (2003), 1185. doi: 10.5802/aif.1977.

[46]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, \emph{Acta Math.}, 13 (1890), 1.

[47]

A. Pronin and D. V. Treschev, Continuous averaging in multi-frequency slow-fast systems,, \emph{Regul. Chaotic Dyn.}, 5 (2000), 157. doi: 10.1070/rd2000v005n02ABEH000138.

[48]

M. Rudnev and S. Wiggins, On a homoclinic splitting problem,, \emph{Regul. Chaotic Dyn.}, 5 (2000), 227. doi: 10.1070/rd2000v005n02ABEH000146.

[49]

D. Sauzin, A new method for measuring the splitting of invariant manifolds,, \emph{Ann. Sci. École Norm. Sup. (4)}, 34 (2001), 159. doi: 10.1016/S0012-9593(00)01063-6.

[50]

C. Simó, Averaging under fast quasiperiodic forcing,, in \emph{Hamiltonian Mechanics} (ed. J. Seimenis) (Toruń, (1993), 13.

[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,, \emph{Nonlinearity}, 14 (2001), 1707. doi: 10.1088/0951-7715/14/6/316.

[52]

D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point,, \emph{Russian J. Math. Phys.}, 5 (1997), 63.

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Amadeu Delshams, Vassili Gelfreich, Angel Jorba and Tere M. Seara. Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing. Electronic Research Announcements, 1997, 3: 1-10.

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