Limit cycles in an m-piecewise discontinuous polynomial differential system

In this paper, I study a planar m-piecewise discontinuous polynomial differential system x = y, y = -x - epsilon(f(x,y) + gm(x, y)h(x)), which has a linear center in each zone partitioned by those switching lines, where f(x, y) = Zni+j=0 ai jxiyj , h(x) = Zlj=0 bjxj, ai j ,bj is an element of R, n, l is an element of N, and gm(x, y) with the positive even number m as the union of m/2 different straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2 pi/m. Using the averaging theory, I provide the lower bound Lm(n, l) for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system.