Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations
type: Item
instance of: scholarly article
title: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations
zbMATH Open document ID: 1390.34082
author: Jihua Yang
author: Liqin Zhao
published in: Journal of Differential Equations
publication date: 2018-02-27T00:00:00Z
review text: The authors consider bifurcation of limit cycles from a period annulus of four particular systems. These four systems have been already studied under continuous polynomial perturbations in previous works. The authors consider discontinuous polynomial perturbations and improve some of the previous results. To prove the existence of such limit cycles, the authors use the maximum number of simple zeros of the first order Melnikov function, the Chebyshev criterion and deduce a system of Picard--Fuchs equations. \newline Firstly, the authors consider the following two quadratic systems of genus one \[ \dot{x} \, = \, y-12x^2+16y^2, \quad \dot{y} \, = \, -x-16xy,\tag{1} \] \[ \dot{x} \, = \, y+4x^2, \quad \dot{y} \, = \, -x+16xy, \tag{2} \] which have been studied (under continuous polynomial perturbations) in [\textit{X. Hong} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 26, No. 2, Article ID 1650020, 16 p. (2016; Zbl 1334.34080)] and [\textit{X. Hong} et al., J. Math. Anal. Appl. 429, No. 2, 924--941 (2015; Zbl 1326.34063)]. The authors consider the following discontinuous perturbation for system (1). Let \(0<|\varepsilon|<<1\) and \(p^{\pm}(x,y)\) and \(q^{\pm}(x,y)\) be real polynomials of degree \(n\). Then, consider \[ \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{cases} \begin{pmatrix} y-12x^2+16y^2 + \varepsilon p^{+}(x,y), \\ -x-16xy + \varepsilon q^{+}(x,y) \end{pmatrix}, \quad & y>0, \\ \begin{pmatrix} y-12x^2+16y^2 + \varepsilon p^{-}(x,y), \\ \displaystyle -x-16xy + \varepsilon q^{-}(x,y) \end{pmatrix}, \quad & y<0. \end{cases} \] One of the main results of the authors is that, by using the first order Melnikov function in \(\varepsilon\), the upper bound of number of limit cycles for this system bifurcating from the period annulus is \(4n - 3\), \((n \geq 4)\), (counting multiplicities). By an analogous perturbation for system (2) the authors obtain an upper bound of \(4n + 3\), \((n \geq 3)\), (counting multiplicities). \newline Then, the a…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6844160
zbMATH Keywords: limit cycle
zbMATH Keywords: discontinuous differential system
zbMATH Keywords: Picard--Fuchs equation
zbMATH Keywords: piecewise smooth differential system
MaRDI profile type: Publication
full work available at URL: j.jde.2018.01.017
OpenAlex ID: W2791133188
cites work: Q5594821
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DOI: 10.1016/J.JDE.2018.01.017
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description: scientific article; zbMATH DE number 6844160