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Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

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    * Corresponding author 
The first author is supported by the grant CSC #201906240094 from the P.R. China. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911
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  • In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center $ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $ with $ m\ge0 $ under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree $ n $ with the discontinuity set $ \{(x, y)\in\mathbb{R}^2: xy = 0\} $. Using the averaging theory up to any order $ N $, we give upper bounds for the maximum number of limit cycles in the function of $ m, n, N $. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.

    Mathematics Subject Classification: Primary: 34C29, 34C25; Secondary: 34C05.

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