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Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing


Authors: Amadeu Delshams, Vassili Gelfreich, Àngel Jorba and Tere M. Seara
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 1-10
MSC (1991): Primary 34C37, 58F27, 58F36; Secondary 11J25
Published electronically: March 12, 1997
MathSciNet review: 1433179
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Abstract: Quasiperiodic perturbations with two frequencies $(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered, where $\gamma $ is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for $\varepsilon $ small enough. The value of the splitting, that turns out to be $\operatorname {O}\left (\exp \left(-\operatorname {const}/\sqrt {\varepsilon }\right )\right )$, is correctly predicted by the Melnikov function.


References [Enhancements On Off] (What's this?)

  • [BCG95] G. Benettin, A. Carati, and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants, Preprint, August 1995.
  • [Ben96] G. Benettin, On the Landau-Teller approximation for adiabatic invariants, In [Sim97].
  • [CG94] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 1, 144 (English, with English and French summaries). MR 1259103 (95b:58056)
  • [DGJS96a] A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Math. Preprints Series 199, Univ. Barcelona, Barcelona, 1996.
  • [DGJS96b] A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara, Splitting of separatrices for (fast) quasiperiodic forcing, In [Sim97].
  • [DS92] Amadeo Delshams and Teresa M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150 (1992), no. 3, 433–463. MR 1204314 (93i:34084)
  • [Gal94] Giovanni 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), no. 3, 343–411. MR 1305589 (97a:58163), http://dx.doi.org/10.1142/S0129055X9400016X
  • [Gel93] V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, Seminar on Dynamical Systems (St.\ Petersburg, 1991) Progr. Nonlinear Differential Equations Appl., vol. 12, Birkhäuser, Basel, 1994, pp. 47–67. MR 1279388 (95f:34056)
  • [Laz84] V. F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, Preprint VINITI No. 6372-84 (in Russian), 1984.
  • [Nei84] A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh. 48 (1984), no. 2, 197–204 (Russian); English transl., J. Appl. Math. Mech. 48 (1984), no. 2, 133–139 (1985). MR 802878 (86j:34043), http://dx.doi.org/10.1016/0021-8928(84)90078-9
  • [Sim94] Carles Simó, Averaging under fast quasiperiodic forcing, Hamiltonian mechanics (Toruń, 1993) NATO Adv. Sci. Inst. Ser. B Phys., vol. 331, Plenum, New York, 1994, pp. 13–34. MR 1316666 (96b:34065)
  • [Sim97] C. Simó (ed.), Hamiltonian systems with three or more degrees of freedom, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., held in S'Agaró, Spain, 19-30 June 1995, Kluwer Acad. Publ., Dordrecht, to appear in 1997.

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Additional Information

Amadeu Delshams
Affiliation: Departament de Matemàtica Aplicada I\ Universitat Politècnica de Catalunya\ Diagonal 647, 08028 Barcelona, Spain
Email: amadeu@ma1.upc.es

Vassili Gelfreich
Affiliation: Departament de Matemàtica Aplicada i Anàlisi\ Universitat de Barcelona\ Gran via 585, 08007 Barcelona, Spain
Address at time of publication: Chair of Applied Mathematics\ St.Petersburg Academy of Aerospace Instrumentation\ Bolshaya Morskaya 67, 190000, St. Petersburg, Russia
Email: gelf@maia.ub.es, gelf@misha.usr.saai.ru

Àngel Jorba
Affiliation: Departament de Matemàtica Aplicada I\ Universitat Politècnica de Catalunya\ Diagonal 647, 08028 Barcelona, Spain
Email: angel@tere.upc.es

Tere M. Seara
Affiliation: Departament de Matemàtica Aplicada I\ Universitat Politècnica de Catalunya\ Diagonal 647, 08028 Barcelona, Spain
Email: tere@ma1.upc.es

DOI: http://dx.doi.org/10.1090/S1079-6762-97-00017-6
PII: S 1079-6762(97)00017-6
Keywords: Splitting of separatrices, quasiperiodic forcing, homoclinic orbits, normal forms.
Received by editor(s): July 9, 1996
Published electronically: March 12, 1997
Communicated by: Jeff Xia
Article copyright: © Copyright 1997 American Mathematical Society