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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

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

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Author(s):
Cardoso, Joao L. [1] ; Llibre, Jaume [2] ; Novaes, Douglas D. [3] ; Tonon, Durval J. [1]
Total Authors: 4
Affiliation:
[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
Total Affiliations: 3
Document type: Journal article
Source: DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL; v. 35, n. 3 FEB 2020.
Web of Science Citations: 0
Abstract

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)

FAPESP's process: 18/16430-8 - Global dynamics of nonsmooth differential equations
Grantee:Douglas Duarte Novaes
Support Opportunities: Regular Research Grants
FAPESP's process: 18/13481-0 - Geometry of control, dynamical and stochastic systems
Grantee:Marco Antônio Teixeira
Support Opportunities: Research Projects - Thematic Grants
FAPESP's process: 19/10269-3 - Ergodic and qualitative theories of dynamical systems II
Grantee:Claudio Aguinaldo Buzzi
Support Opportunities: Research Projects - Thematic Grants