Geodesic
Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing
Delshams, Amadeu1 ; Gelfreich, Vassili2, 3 ; Jorba, Àngel1 ; Seara, Tere1
1 Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya Diagonal 647, 08028 Barcelona, Spain
2 Departament de Matemàtica Aplicada i Anàlisi Universitat de Barcelona Gran via 585, 08007 Barcelona, Spain
3 Chair of Applied Mathematics St.Petersburg Academy of Aerospace Instrumentation Bolshaya Morskaya 67, 190000, St. Petersburg, Russia
Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 1-10 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S1079-6762-97-00017-6

Delshams, Amadeu 1 ; Gelfreich, Vassili 2, 3 ; Jorba, Àngel 1 ; Seara, Tere 1

1 Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya Diagonal 647, 08028 Barcelona, Spain
2 Departament de Matemàtica Aplicada i Anàlisi Universitat de Barcelona Gran via 585, 08007 Barcelona, Spain
3 Chair of Applied Mathematics St.Petersburg Academy of Aerospace Instrumentation Bolshaya Morskaya 67, 190000, St. Petersburg, Russia 
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Delshams, Amadeu; Gelfreich, Vassili; Jorba, Àngel; Seara, Tere. Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing. Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 1-10. doi: 10.1090/S1079-6762-97-00017-6

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