Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line

Autor(es)

Carmona, Victoriano ; Fernandez-Sanchez, Fernando ; Novaes, Douglas D.

• Tipo de documento: Artigo Científico
• Fonte: Applied Mathematics Letters; v. 137, p. 8-pg., 2023-03-01.

Resumo

The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number L* <= 8 for which any planar piecewise linear differential system with two zones separated by a straight line has no more than L* limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincare half-maps for linear differential systems with an extension of Khovanskii's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields. (c) 2022 Elsevier Ltd. All rights reserved. (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