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Exponentially small splitting of separatrices under fast quasiperiodic forcing
Delshams Valdés, Amadeu; Gelfreich, Vassili; Jorba, Angel; Martínez-Seara Alonso, M. Teresa
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We consider fast quasiperiodic perturbations with two frequencies $(1/\varepsilon,\gamma/\varepsilon)$ of a pendulum, where $\gamma$ is the golden mean number. The complete system has a two-dimensional invariant torus in a neighbourhood of the saddle point. We study the splitting of the three-dimensional invariant manifolds associated to this torus. Provided that the perturbation amplitude is small enough with respect to $\varepsilon $, and some of its Fourier coefficients (the ones associated to Fibonacci numbers), are separated from zero, it is proved that the invariant manifolds split and that the value of the splitting, which turns out to be exponentially small with respect to $\varepsilon $, is correctly predicted by the Melnikov function.
Global analysis (Mathematics)
Hamiltonian dynamical systems
Lagrangian functions
Nonlinear Dynamics
Differential equations
quasiperiodic forcing
Varietats (Matemàtica)
Hamilton, Sistemes de
Lagrange, Funcions de
Partícules (Física nuclear)
Equacions diferencials ordinàries
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::58 Global analysis, analysis on manifolds
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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