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Limit cycles in some piecewise differential systems

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)

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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)
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, . (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, . (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, . (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, . (17/03352-6, 19/10269-3, 13/24541-0)
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, . (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, . (17/03352-6, 13/24541-0)

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