Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations
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 authors consider two of the four Loud quadratic isochronous centers \[ \dot{x} \, = \, -y+x^2-y^2, \quad \dot{y} \, = \, x + 2xy, \tag{3} \] \[ \dot{x} \, = \, -y+x^2, \quad \dot{y} \, = \, x+xy. \tag{4} \] The bifurcation of limit cycles from the period annulus surrounding the origin of these systems under continuous quadratic perturbations have been studied by \textit{C. Chicone} and \textit{M. Jacobs} [J. Differ. Equations 91, No. 2, 268--326 (1991; Zbl 0733.34045)] and at most two limit cycles have been found. The perturbation inside a class of piecewise smooth quadratic polynomial differential systems have been done by \textit{J. Llibre} and \textit{A. C. Mereu} [J. Math. Anal. Appl. 413, No. 2, 763--775 (2014; Zbl 1318.34049)] and at most \(4\) and \(5\) limit cycles have been found for systems (3) and (4), respectively, by the averaging method at first order. Here the authors consider the following perturbations of system (3). Let \(0<|\varepsilon|<<1\) and \(p^{\pm}_2(x,y)\) and \(q^{\pm}_2(x,y)\) be real polynomials of degree \(2\). Then, consider \[ \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}= \begin{cases} \begin{pmatrix} -y+x^2-y^2 + \varepsilon p^{+}_2(x,y), \\ x + 2xy + \varepsilon q^{+}_2(x,y) \end{pmatrix}, \quad & y>0, \\ \begin{pmatrix} -y+x^2-y^2 + \varepsilon p^{-}_2(x,y), \\ x + 2xy + \varepsilon q^{-}_2(x,y) \end{pmatrix}, \quad & y<0. \end{cases} \] The second main result 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 \(5\) (counting multiplicities). An analogous perturbation for system (4) leads to an upper bound of \(6\) limit cycles (counting multiplicities).
scientific article; zbMATH DE number 6844160
scientific article