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Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

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Abstract

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.

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Authors and Affiliations

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

    Marcel Guardia & Tere M. Seara

  2. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avda Països Catalans 26, 43007, Tarragona, Spain

    Carme Olivé

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  1. Marcel Guardia
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  2. Carme Olivé
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  3. Tere M. Seara
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Correspondence to Tere M. Seara.

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Communicated by J. Marsden.

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Guardia, M., Olivé, C. & Seara, T.M. Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited. J Nonlinear Sci 20, 595–685 (2010). https://doi.org/10.1007/s00332-010-9068-8

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