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The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

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Abstract

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

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Acknowledgements

This work is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación Grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911

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Authors and Affiliations

  1. Département de Mathématiques, Université Mohamed El Bachir El Ibrahimi, 34000, El Anasser, Bordj Bou Arréridj, Algeria

    Rebiha Benterki

  2. Departament de Matematiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

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  1. Rebiha Benterki
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  2. Jaume Llibre
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Correspondence to Rebiha Benterki.

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Benterki, R., Llibre, J. The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems. Nonlinear Dyn 102, 2453–2466 (2020). https://doi.org/10.1007/s11071-020-06045-z

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  • DOI: https://doi.org/10.1007/s11071-020-06045-z

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