Limit Cycles of Discontinuous Perturbed Quadratic Center via the Second Order Averaging Method

In this paper, we study the number of limit cycles for a piecewise smooth quadratic integrable but non-Hamiltonian system having a center, which is separated by a straight line x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document} (called as a switching line). When the quadratic center perturbed inside discontinuous quadratic and cubic homogenous polynomials starting with terms of degree 1, by applying the second order averaging method of discontinuous differential equations, we obtain two criteria on the lower upper bounds of the maximum number of limit cycles bifurcating from the period annulus. And the bounds can be realized.