Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

Jackson Itikawa; Jaume Llibre; Ana Cristina Mereu; Regilene Oliveira

doi:10.3934/dcdsb.2017136

Article Contents

2017, Volume 22, Issue 9: 3259-3272. Doi: 10.3934/dcdsb.2017136

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Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

1.
Departamento de Matemática, ICMC-Universidade de Sãao Paulo, Avenida Trabalhador Sãao-carlense, 400, Sãao Carlos, SP, 13566-590, Brazil
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
3.
Departamento de Física, Química e Matemática, UFSCar, Sorocaba, SP, 18052-780, Brazil

Received: July 31, 2016

Revised: February 28, 2017

Published: October 2017

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Abstract

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems $\dot{x}=-y+x^2, \;\dot{y}=x+xy$, and $\dot{x}=-y+x^2y, \;\dot{y}=x+xy^2$, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively.

Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.

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Mathematics Subject Classification: Primary:34A36, 34C07, 34C25, 34C29, 37G15.

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Table 1. Number of limit cycles for continuous and discontinuous quadratic and cubic differential systems

Case Number of limit cycles for

system (1) system (2)

Continuous 2 3

Discontinuous with 2 zones 5 7

Discontinuous with 4 zones 10 12

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