Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x(center dot)=y,y(center dot)=-x-epsilon<middle dot>(f(x)<middle dot>y+sgn(y)<middle dot>g(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +\textrm{sgn}(y)\cdot g(x))$$\end{document}. Using the averaging method, we were able to generalize a previous result for Li & eacute;nard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of |epsilon|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\varepsilon }|$$\end{document}, the number hm,n=n2+m2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}=\left[ \frac{n}{2}\right] +\left[ \frac{m}{2}\right] +1$$\end{document} serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center x(center dot)=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(center dot)=-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}. Furthermore, we demonstrate that it is indeed possible to obtain a system with hm,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}$$\end{document} limit cycles.