Journal of Nonlinear Science

, Volume 20, Issue 5, pp 595–685 | Cite as

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

  • Marcel Guardia
  • Carme Olivé
  • Tere M. SearaEmail author


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.


Exponentially small splitting of separatrices Melnikov method Resurgence theory Averaging Complex matching 

Mathematics Subject Classification (2000)

34C29 34C37 37C29 34E10 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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