On the Equivalence of the Melnikov Functions Method and the Averaging Method

• Adriana Buica ^[1]
  1. [1] Universitatea BabesBolyai (Rumanía)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 16, Nº 3, 2017, págs. 547-560
• Idioma: inglés
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• Resumen

  • We prove the equivalence between the Melnikov functions method and the averaging method as tools for finding limit cycles of analytic planar differential systems which are perturbations of a period annulus. We consider any possible change of variables to transform the planar system into a scalar periodic equation which perturbs a continuum of constant solutions. We prove that the Poincaré return map of the planar system and the Poincaré translation map of the scalar equation coincide. For distinct specific changes of variables this was stated before in 2004 by Buic˘a–Llibre and proved in 2015 by Han–Romanovski–Zhang.