Averaging under Fast Quasiperiodic Forcing

  • Carles Simó
Part of the NATO ASI Series book series (NSSB, volume 331)


We consider a non autonomous system of ordinary differential equations. Assume that the time dependence is quasiperiodic with large basic frequencies, ω/ε and that the ω vector satisfies a diophantine condition. Under suitable hypothesis of analyticity, there exists an analytic (time depending) change of coordinates, such that the new vector field is the sum of an autonomous part and a time dependent remainder. The remainder has an exponentially small bound of the form exp(− a ), where c and a are positive constants. The proof is obtained by iteration of an averaging process. An application is made to the splitting of the separatrices of a two-dimensional normally hyperbolic torus, including several aspects: formal approximation of the torus and their invariant manifolds, numerical computations of the splitting, a first order analysis using a Melnikov approach and the bifurcations of the set of homoclinic orbits.


Invariant Manifold Homoclinic Orbit Invariant Torus Moderate Time Interval Melnikov Function 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Carles Simó
    • 1
  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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