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Piecewise linear differential systems with only centers can create limit cycles?

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Abstract

In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.

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Acknowledgements

We thank the reviewers for their comments which help us to improve this paper. The first author is partially supported by a FEDER-MINECO Grant MTM2016-77278-P, a MINECO Grant MTM2013-40998-P, an AGAUR Grant Number 2014SGR-568, and a CAPES Grant 88881.030454/ 2013-01 do Programa CSF-PVE. The second author is partially supported by FAPESP under Grant Number 2012/18780-0.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemàtica, Universidade Estadual de Campinas, Campinas, SP, 13083–970, Brazil

    Marco Antonio Teixeira

Authors
  1. Jaume Llibre
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  2. Marco Antonio Teixeira
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Correspondence to Marco Antonio Teixeira.

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Llibre, J., Teixeira, M.A. Piecewise linear differential systems with only centers can create limit cycles?. Nonlinear Dyn 91, 249–255 (2018). https://doi.org/10.1007/s11071-017-3866-6

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  • DOI: https://doi.org/10.1007/s11071-017-3866-6

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