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The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

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Abstract

We provide the maximum number of limit cycles for discontinuous piecewise differential systems separated by a straight line and formed by a linear Hamiltonian saddle and one of the four families of quadratic isochronous centers. Hence we have solved the extension of the 16th Hilbert problem to such piecewise differential systems and for two of these four classes of discontinuous piecewise differential systems the obtained upper bound for the maximum number of limit cycles is reached.

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Funding

Jaume Llibre is partially supported by the Agencia Estatal de Investigación grant PID2022-136613NB-100, the H2020 European Research Council grant MSCA-RISE-2017-777911, AGAUR (Generalitat de Catalunya) grant 2021SGR00113, and by the Acadèmia de Ciències i Arts de Barcelona. Claudia Valls 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, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, 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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Llibre, J., Valls, C. The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line. Differ Equ Dyn Syst (2024). https://doi.org/10.1007/s12591-024-00695-w

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