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The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Catalonia, Spain

E-mail Address: jaume.llibre@uab.cat

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and

Claudia Valls

https://orcid.org/0000-0001-8279-1229

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

E-mail Address: cvalls@math.ist.utl.pt

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https://doi.org/10.1142/S0218127423501961Cited by:2 (Source: Crossref)

Abstract

We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type ${I I}_{2}$ and limit cycles of type ${I I}_{4}$ , respectively. We prove that the maximum numbers of limit cycles of types ${I I}_{2}$ and ${I I}_{4}$ are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.

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