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

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Abstract

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 Buică–Llibre and proved in 2015 by Han–Romanovski–Zhang.

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Acknowledgments

This work was supported by UBB grant GSCE-30255-2015. The author thanks Maite Grau for useful discussions, especially with respect to Theorem 7, and Hector Giacomini for a question during a conference few years ago which led to the writing of this paper. We are also grateful to the reviewer for his comments which helped to improve the presentation of this paper.

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  1. Departamentul de Matematică, Universitatea BabeşBolyai, Str. Kogălniceanu 1, 400084, Cluj-Napoca, Romania

    Adriana Buică

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  1. Adriana Buică
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Correspondence to Adriana Buică.

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Buică, A. On the Equivalence of the Melnikov Functions Method and the Averaging Method. Qual. Theory Dyn. Syst. 16, 547–560 (2017). https://doi.org/10.1007/s12346-016-0216-x

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  • DOI: https://doi.org/10.1007/s12346-016-0216-x

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