Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

Autor(es)

Cardoso, Joao L. ^[1] ; Llibre, Jaume ^[2] ; Novaes, Douglas D. ^[3] ; Tonon, Durval J. ^[1]

Afiliação do(s) autor(es)

^[1] Univ Fed Goias, Inst Math & Stat, Goiania, Go - Brazil

^[2] Univ Autonoma Barcelona, Dept Matemat, Barcelona, Catalonia - Spain

^[3] Univ Estadual Campinas, Dept Matemat, Sao Paulo - Brazil

• Tipo de documento: Artigo Científico
• Fonte: DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL; v. 35, n. 3 FEB 2020.
• Citações Web of Science: 0

Resumo

In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for yy>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation. (AU)

• Processo FAPESP: 18/16430-8 - Dinâmica global das equações diferenciais não suaves
• Beneficiário: Douglas Duarte Novaes
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• 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: 19/10269-3 - Teorias ergódica e qualitativa dos sistemas dinâmicos II
• Beneficiário: Claudio Aguinaldo Buzzi
• Modalidade de apoio: Auxílio à Pesquisa - Temático