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Crossing limit cycles for discontinuous piecewise linear differential centers separated by three parallel straight lines

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Abstract

In this paper we study the continuous and discontinuous planar piecewise differential systems formed by four linear centers separated by three parallel straight lines denoted by \(\Sigma =\{(x,y)\in {\mathbb {R}}^2: x=-p, x=0, x=q,\ p,q>0\}.\) We prove that when these piecewise differential systems are continuous they have no limit cycles. While for the discontinuous case we show that they can have at most four limit cycles and we also provide examples of such systems with zero, one, and two limit cycles. In particular we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems.

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Acknowledgements

The second author is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación Grants PID2019-104658GB-I00, and the H2020 European Research Council Grant MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2019. The present paper is part of the thesis of the first author.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad del Bío-Bío, Avda. Collao, 1202, Concepción, Chile

    Maria Elisa Anacleto & Claudio Vidal

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

    Jaume Llibre

  3. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Claudia Valls

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  1. Maria Elisa Anacleto
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  2. Jaume Llibre
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  3. Claudia Valls
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  4. Claudio Vidal
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Correspondence to Maria Elisa Anacleto.

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Anacleto, M.E., Llibre, J., Valls, C. et al. Crossing limit cycles for discontinuous piecewise linear differential centers separated by three parallel straight lines. Rend. Circ. Mat. Palermo, II. Ser 72, 1739–1750 (2023). https://doi.org/10.1007/s12215-022-00766-3

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  • DOI: https://doi.org/10.1007/s12215-022-00766-3

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