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The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve \(y=x^{n}\)

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Abstract

We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve \(y=x^n\) with \(n \ge 2\). We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of n, proving the extended 16th Hilbert problem in this case. In particular, we show that for \(n=2\) this bound can be reached.

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Acknowledgements

We thank to the reviewers their comments that help us to improve the presentation of our results. The first author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, the H2020 European Research Council grant MSCA-RISE-2017-777911, AGAUR (Generalitat de Catalunya) grant 2022-SGR 00113, and by the Acadèmia de Ciències i Arts de Barcelona. The second author is partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

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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 Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001, Lisboa, Portugal

    Claudia Valls

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  1. Jaume Llibre
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  2. Claudia Valls
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Correspondence to Claudia Valls.

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Communicated by Lucia Di Vizio.

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Llibre, J., Valls, C. The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve \(y=x^{n}\). Math Phys Anal Geom 26, 25 (2023). https://doi.org/10.1007/s11040-023-09467-4

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