Limit cycles in some piecewise differential systems

• Grant number: 17/03352-6
• Support type: Research Grants - Visiting Researcher Grant - International
• Duration: May 15, 2017 - June 22, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal researcher: Claudio Aguinaldo Buzzi
• Grantee: Claudio Aguinaldo Buzzi
• Visiting researcher: Joan Torregrosa i Arús
• Visiting researcher institution: Universitat Autònoma de Barcelona (UAB), Spain
• Home Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

We will study the isolated periodic orbits (limit cycles) in planar piecewise polynomial systems. The first problem is defined in four zones defined by two straight lines that crosses perpendicularly. Here we consider two or four different linear systems. We prove that in a symmetric case (two systems) there exist systems with five limit cycles bifurcating far from the origin and in the non-symmetric case (four systems) the number increase to ten. The study of the Poincaré return map, computing the well known Lyapunov quantities, is the main technique in this problem. The second problem is defined in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n, no more than Nn - 1 limit cycles appear up to a study of order N. We also show that this upper bound is reached for orders one and two. The Poincaré-Pontryagin-Melnikov theory is the main technique used to prove the results in this second problem. (AU)

Scientific publications

(6)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

JIMENEZ, JEIDY J.; LLIBRE, JAUME; MEDRADO, JOAO C.. Crossing limit cycles for piecewise linear differential centers separated by a reducible cubic curve. Electronic Journal of Qualitative Theory of Differential Equations, n. 19, p. 1-48, 2020. (17/03352-6, 13/24541-0)

JIMENEZ, JOHANA; LLIBRE, JAUME; MEDRADO, JOAO C.. CROSSING LIMIT CYCLES FOR A CLASS OF PIECEWISE LINEAR DIFFERENTIAL CENTERS SEPARATED BY A CONIC. Electronic Journal of Differential Equations, MAY 7 2020. (17/03352-6, 13/24541-0)

BUZZI, CLAUDIO A.; SILVA LIMA, MAURICIO FIRMINO; TORREGROSA, JOAN. Limit cycles via higher order perturbations for some piecewise differential systems. PHYSICA D-NONLINEAR PHENOMENA, v. 371, p. 28-47, MAY 15 2018. (17/03352-6, 12/18780-0, 13/24541-0)

BUZZI, CLAUDIO A.; GASULL, ARMENGOL; TORREGROSA, JOAN. Algebraic Limit Cycles in Piecewise Linear Differential Systems. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 28, n. 3, MAR 2018. (17/03352-6, 13/24541-0)

BUZZI, CLAUDIO A.; MEDRADO, JOAO C.; TORREGROSA, JOAN. Limit cycles in 4-star-symmetric planar piecewise linear systems. Journal of Differential Equations, v. 268, n. 5, p. 2414-2434, FEB 15 2020. (17/03352-6, 13/24541-0)

BASTOS, JEFFERSON L. R.; BUZZI, CLAUDIO A.; TORREGROSA, JOAN. ORBITALLY SYMMETRIC SYSTEMS WITH APPLICATIONS TO PLANAR CENTERS. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, v. 20, n. 10, p. 3301-3329, OCT 2021. (17/03352-6, 19/10269-3, 13/24541-0)