Journal of Nonlinear Science

, Volume 20, Issue 5, pp 595–685

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited



In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.


Exponentially small splitting of separatrices Melnikov method Resurgence theory Averaging Complex matching 

Mathematics Subject Classification (2000)

34C29 34C37 37C29 34E10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1992). Reprint of the 1972 edition Google Scholar
  2. Angenent, S.: A variational interpretation of Mel’nikov’s function and exponentially small separatrix splitting. In: Symplectic Geometry. London Math. Soc. Lecture Note Ser., vol. 192, pp. 5–35. Cambridge Univ. Press, Cambridge (1993) Google Scholar
  3. Baldomá, I.: The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems. Nonlinearity 19(6), 1415–1445 (2006) MATHCrossRefMathSciNetGoogle Scholar
  4. Baldomá, I., Fontich, E.: Exponentially small splitting of invariant manifolds of parabolic points. Mem. Am. Math. Soc. 167(792), x+83 (2004) Google Scholar
  5. Baldomá, I., Fontich, E.: Exponentially small splitting of separatrices in a weakly hyperbolic case. J. Differ. Equ. 210(1), 106–134 (2005) MATHCrossRefGoogle Scholar
  6. Baldomá, I., Seara, T.M.: Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity. J. Nonlinear Sci. 16(6), 543–582 (2006) MATHCrossRefMathSciNetGoogle Scholar
  7. Baldomá, I., Seara, T.M.: The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 323–347 (2008) MATHMathSciNetGoogle Scholar
  8. Balser, W.: From Divergent Power Series to Analytic Functions. Lecture Notes in Mathematics, vol. 1582. Springer, Berlin (1994). Theory and application of multisummable power series MATHGoogle Scholar
  9. Benseny, A., Olivé, C.: High precision angles between invariant manifolds for rapidly forced Hamiltonian systems. In: Proceedings Equadiff91, pp. 315–319 (1993) Google Scholar
  10. Bonet, C., Sauzin, D., Seara, T., València, M.: Adiabatic invariant of the harmonic oscillator, complex matching and resurgence. SIAM J. Math. Anal. 29(6), 1335–1360 (1998) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  11. Candelpergher, B., Nosmas, J.-C., Pham, F.: Approche de la résurgence. Actualités Mathématiques [Current Mathematical Topics]. Hermann, Paris (1993) Google Scholar
  12. Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor. 60(1), 144 (1994) MathSciNetGoogle Scholar
  13. Delshams, A., Gutiérrez, P.: Splitting potential and the Poincaré–Melnikov method for whiskered tori in Hamiltonian systems. J. Nonlinear Sci. 10(4), 433–476 (2000) MATHCrossRefMathSciNetGoogle Scholar
  14. Delshams, A., Ramírez-Ros, R.: Melnikov potential for exact symplectic maps. Commun. Math. Phys. 190(1), 213–245 (1997) MATHCrossRefGoogle Scholar
  15. Delshams, A., Ramírez-Ros, R.: Exponentially small splitting of separatrices for perturbed integrable standard-like maps. J. Nonlinear Sci. 8(3), 317–352 (1998) MATHCrossRefMathSciNetGoogle Scholar
  16. Delshams, A., Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Commun. Math. Phys. 150(3), 433–463 (1992) MATHCrossRefMathSciNetGoogle Scholar
  17. Delshams, A., Seara, T.M.: Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J., 3, Paper 4, 40 pp. (electronic) (1997) Google Scholar
  18. Delshams, A., Gelfreich, V., Jorba, À., Seara, T.M.: Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys. 189(1), 35–71 (1997) MATHCrossRefMathSciNetGoogle Scholar
  19. Delshams, A., Gutiérrez, P., Seara, T.M.: Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete Contin. Dyn. Syst. 11(4), 785–826 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. Écalle, J.: Les fonctions résurgentes. Tome I. Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 5. Université de Paris-Sud Département de Mathématique, Orsay (1981a). Les algèbres de fonctions résurgentes. [The algebras of resurgent functions], With an English foreword Google Scholar
  21. Écalle, J.: Les fonctions résurgentes. Tome II. Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 6. Université de Paris-Sud Département de Mathématique, Orsay (1981b). Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration] Google Scholar
  22. Eliasson, L.H.: Biasymptotic solutions of perturbed integrable Hamiltonian systems. Bol. Soc. Brasil. Mat. (N.S.) 25(1), 57–76 (1994) MATHCrossRefMathSciNetGoogle Scholar
  23. Ellison, J.A., Kummer, M., Sáenz, A.W.: Transcendentally small transversality in the rapidly forced pendulum. J. Dyn. Differ. Equ. 5(2), 241–277 (1993) MATHCrossRefGoogle Scholar
  24. Fiedler, B., Scheurle, J.: Discretization of homoclinic orbits, rapid forcing and “invisible” chaos. Mem. Am. Math. Soc. 119(570), viii+79 (1996) Google Scholar
  25. Fontich, E.: Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations. Nonlinear Anal. 20(6), 733–744 (1993) MATHCrossRefMathSciNetGoogle Scholar
  26. Fontich, E.: Rapidly forced planar vector fields and splitting of separatrices. J. Differ. Equ. 119(2), 310–335 (1995) MATHCrossRefMathSciNetGoogle Scholar
  27. Fontich, E., Simó, C.: The splitting of separatrices for analytic diffeomorphisms. Ergod. Theory Dyn. Syst. 10(2), 295–318 (1990) MATHGoogle Scholar
  28. Gallavotti, G.: 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(3), 343–411 (1994) MATHCrossRefMathSciNetGoogle Scholar
  29. Gallavotti, G., Gentile, G., Mastropietro, V.: Separatrix splitting for systems with three time scales. Commun. Math. Phys. 202(1), 197–236 (1999) MATHCrossRefMathSciNetGoogle Scholar
  30. Gelfreich, V.G.: Separatrices splitting for the rapidly forced pendulum. In: Seminar on Dynamical Systems, St. Petersburg, 1991. Progr. Nonlinear Differential Equations Appl., vol. 12, pp. 47–67. Birkhäuser, Basel (1994) Google Scholar
  31. Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Physica D 101(3–4), 227–248 (1997a) MATHCrossRefMathSciNetGoogle Scholar
  32. Gelfreich, V.G.: Reference systems for splittings of separatrices. Nonlinearity 10(1), 175–193 (1997b) MATHCrossRefMathSciNetGoogle Scholar
  33. Gelfreich, V.G.: A proof of the exponentially small transversality of the separatrices for the standard map. Commun. Math. Phys. 201(1), 155–216 (1999) MATHCrossRefMathSciNetGoogle Scholar
  34. Gelfreich, V.G.: Separatrix splitting for a high-frequency perturbation of the pendulum. Russ. J. Math. Phys. 7(1), 48–71 (2000) MATHMathSciNetGoogle Scholar
  35. Gelfreich, V.G., Naudot, V.: Width of homoclinic zone in the parameter space for quadratic maps. Exp. Math. 18(4), 409–427 (2009) MATHMathSciNetGoogle Scholar
  36. Gelfreich, V., Sauzin, D.: Borel summation and splitting of separatrices for the Hénon map. Ann. Inst. Fourier (Grenoble) 51(2), 513–567 (2001) MATHMathSciNetGoogle Scholar
  37. Gelfreich, V.G., Lazutkin, V.F., Tabanov, M.B.: Exponentially small splittings in Hamiltonian systems. Chaos 1(2), 137–142 (1991) MATHCrossRefMathSciNetGoogle Scholar
  38. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983) MATHGoogle Scholar
  39. Holmes, P.J., Marsden, J.E.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Ration. Mech. Anal. 76(2), 135–165 (1981) MATHCrossRefMathSciNetGoogle Scholar
  40. Holmes, P.J., Marsden, J.E.: Melnikov’s method and Arnol’d diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23(4), 669–675 (1982) MATHCrossRefMathSciNetGoogle Scholar
  41. Holmes, P.J., Marsden, J.E.: Horseshoes and Arnol’d diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32(2), 273–309 (1983) MATHCrossRefMathSciNetGoogle Scholar
  42. Holmes, P.J., Marsden, J.E., Scheurle, J.: Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In: Hamiltonian Dynamical Systems. Contemp. Math., vol. 81 (1988) Google Scholar
  43. Lazutkin, V.F.: Splitting of separatrices for the Chirikov standard map. VINITI 6372/82, 1984. Preprint (Russian) Google Scholar
  44. Lazutkin, V.F.: Splitting of separatrices for the Chirikov standard map. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (Teor. Predst. Din. Sist. Spets. Vyp. 8), 25–55, 285 (2003) Google Scholar
  45. Lochak, P., Marco, J.-P., Sauzin, D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Mem. Am. Math. Soc. 163(775), viii+145 (2003) Google Scholar
  46. Lombardi, E.: Oscillatory Integrals and Phenomena Beyond All Algebraic Orders. Lecture Notes in Mathematics, vol. 1741. Springer, Berlin (2000). With applications to homoclinic orbits in reversible systems MATHGoogle Scholar
  47. Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Mosc. Math. Soc. 12, 1–57 (1963) Google Scholar
  48. Neĭshtadt, A.I.: The separation of motions in systems with rapidly rotating phase. Prikl. Mat. Meh. 48(2), 197–204 (1984) Google Scholar
  49. Olivé, C.: Càlcul de l’escissió de separatrius usant tècniques de matching complex i ressurgència aplicades a l’equació de Hamilton–Jacobi. (2006)
  50. Olivé, C., Sauzin, D., Seara, T.M.: Resurgence in a Hamilton–Jacobi equation. In: Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), vol. 53(4), pp. 1185–1235 (2003) Google Scholar
  51. Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13, 1–270 (1890) Google Scholar
  52. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste, vols. 1, 2, 3. Gauthier-Villars, Paris (1892–1899) Google Scholar
  53. Sanders, J.A.: Melnikov’s method and averaging. Celest. Mech. 28(1–2), 171–181 (1982) MATHGoogle Scholar
  54. Sauzin, D.: Résurgence paramétrique et exponentielle petitesse de l’écart des séparatrices du pendule rapidement forcé. Ann. Inst. Fourier 45(2), 453–511 (1995) MATHMathSciNetGoogle Scholar
  55. Sauzin, D.: A new method for measuring the splitting of invariant manifolds. Ann. Sci. Éc. Norm. Super. (4) 34 (2001) Google Scholar
  56. Scheurle, J.: Chaos in a Rapidly Forced Pendulum Equation. Contemp. Math., vol. 97. Am. Math. Soc., Providence (1989) Google Scholar
  57. Scheurle, J., Marsden, J.E., Holmes, P.J.: Exponentially small estimates for separatrix splittings. In: Asymptotics Beyond All Orders, La Jolla, CA, 1991. NATO Adv. Sci. Inst. Ser. B, Phys., vol. 284, pp. 187–195. Plenum, New York (1991) Google Scholar
  58. Seara, T.M., Villanueva, J.: Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map. Nonlinearity 13(5), 1699–1744 (2000) MATHCrossRefMathSciNetGoogle Scholar
  59. Simó, C.: Averaging under fast quasiperiodic forcing. In: Hamiltonian Mechanics, Toruń, 1993. NATO Adv. Sci. Inst. Ser. B, Phys., vol. 331, pp. 13–34. Plenum, New York (1994) Google Scholar
  60. Smale, S.: Diffeomorphisms with many periodic points. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 63–80. Princeton Univ. Press, Princeton (1965) Google Scholar
  61. Treschev, D.: Hyperbolic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys. 2(1), 93–110 (1994) MathSciNetGoogle Scholar
  62. Treschev, D.: Separatrix splitting for a pendulum with rapidly oscillating suspension point. Russ. J. Math. Phys. 5(1), 63–98 (1997) MATHMathSciNetGoogle Scholar
  63. Treschev, D.: Width of stochastic layers in near-integrable two-dimensional symplectic maps. Physica D 116(1–2), 21–43 (1998) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

Personalised recommendations