Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

Jaume Llibre; Yilei Tang

doi:10.3934/dcdsb.2018236

Article Contents

2019, Volume 24, Issue 4: 1769-1784. Doi: 10.3934/dcdsb.2018236

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Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

Jaume Llibre^1, and
Yilei Tang^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received: October 31, 2017

Revised: January 31, 2018

Early access: August 2018

Published: January 2019

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Abstract

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line.

We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $ n $ for $ n = 1, 2, 3, 4, 5 $. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations.

Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi-homogenous polynomials.

Keywords:

Periodic solution,
limit cycle,
discontinuous piecewise differential system,
averaging theory.

Mathematics Subject Classification: 34C29, 34C25, 34C05.

Citation:

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Figure 1. Existence of closed orbits for system (29)

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Figure 2. Existence of global center for system (29)

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Table 1. Maximum number of limit cycles bifurcating from the periodic orbits of the linear center using averaging theory of order $n$

Order $n$ $L_1(n)$ $L_2(n)$ $L_3(n)$ $L_2^I(n)$ $L_3^I(n)$

1 1 2 3 0 1

2 1 3 5 1 2

3 2 5 8 1 3

4 3 6 11 2 4

5 3 8 13 2 5

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