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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.
Copyright 1997 American Mathematical Society
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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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