On the Number of Limit Cycles Bifurcating from the Linear Center with a Cubic Switching Curve

This paper studies the bifurcations of limit cycles from the system x˙=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{x}}=y$$\end{document}, y˙=-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{y}}=-x$$\end{document} with the switching curve y=x3/3-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y = x^3/3-x$$\end{document} under the perturbations of arbitrary polynomials of x and y with degree n. We obtain the lower bound and upper bound of the maximum number of limit cycles bifurcating from h∈(0,3/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in (0,3/2)$$\end{document} if the first order Melnikov function is not identically 0. When the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first order Melnikov function. We also give an example to illustrate our result.