Nonstationary normal forms and rigidity of group actions

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

Amadeu Delshams, Vassili Gelfreich, Àngel Jorba and Tere M. Seara

Abstract.

Quasiperiodic perturbations with two frequencies $(1/\epsilon, \gamma/\varepsilon)$ 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 $\Or \left(\exp\left(-\const /\sqrt{\varepsilon }\right)\right)$ , is correctly predicted by the Melnikov function.

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Article Info

ERA Amer. Math. Soc. 03 (1997), pp. 1-10
Publisher Identifier: S 1079-6762(97)00017-6
1991 Mathematics Subject Classification. Primary 34C37, 58F27, 58F36; Secondary 11J25
Key words and phrases. Splitting of separatrices, quasiperiodic forcing, homoclinic orbits, normal forms
Received by the editors July 9, 1996
Posted on March 12, 1997
Communicated by Jeff Xia
Comments (When Available)

Amadeu Delshams

Departament de Matemàtica Aplicada I
Universitat Politècnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain

E-mail address: amadeu@ma1.upc.es

Vassili Gelfreich

Departament de Matemàtica Aplicada i Anàlisi
Universitat de Barcelona
Gran via 585, 08007 Barcelona, Spain

Current address: Chair of Applied Mathematics
St. Petersburg Academy of Aerospace Instrumentation
Bolshaya Morskaya 67, 190000, St. Petersburg, Russia

E-mail address: gelf@maia.ub.es, gelf@misha.usr.saai.ru

Àngel Jorba

Departament de Matemàtica Aplicada I
Universitat Politècnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain

E-mail address: angel@tere.upc.es

Tere M. Seara

Departament de Matemàtica Aplicada I
Universitat Politècnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain

E-mail address: tere@ma1.upc.es

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