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Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

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Autor(es):
Carmona, Victoriano ; Fernandez-Sanchez, Fernando ; Novaes, Douglas D.
Número total de Autores: 3
Tipo de documento: Artigo Científico
Fonte: COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION; v. 123, p. 18-pg., 2023-04-28.
Resumo

In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincare half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). (AU)

Processo FAPESP: 19/10269-3 - Teorias ergódica e qualitativa dos sistemas dinâmicos II
Beneficiário:Claudio Aguinaldo Buzzi
Modalidade de apoio: Auxílio à Pesquisa - Temático
Processo FAPESP: 21/10606-0 - Sobre ciclos limites em espaços vetoriais lineares por partes com variedade de descontinuidade algébrica
Beneficiário:Douglas Duarte Novaes
Modalidade de apoio: Bolsas no Exterior - Pesquisa
Processo FAPESP: 18/13481-0 - Geometria de sistemas de controle, sistemas dinâmicos e estocásticos
Beneficiário:Marco Antônio Teixeira
Modalidade de apoio: Auxílio à Pesquisa - Temático
Processo FAPESP: 22/09633-5 - Teoria da média no estudo de toros invariantes e comportamento periódico em equações e inclusões diferenciais
Beneficiário:Douglas Duarte Novaes
Modalidade de apoio: Auxílio à Pesquisa - Regular