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ISSN 1079-6762

 
 

 

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
DOI: https://doi.org/10.1090/S1079-6762-97-00017-6
Published electronically: March 12, 1997
MathSciNet review: 1433179
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Abstract | References | Similar Articles | Additional Information

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 $\mathrm {O} \left (\exp \left (-\mathrm {const}/\sqrt {\varepsilon }\right )\right )$, is correctly predicted by the Melnikov function.


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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

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