Limit cycles in an $m$-piecewise discontinuous polynomial differential system

In this paper, I study a planar $m$-piecewise discontinuous polynomial differential system $\dot{x}=y,\dot{y}=-x-\epsilon (f(x,y)+g_m(x,y)h(x))$, which has a linear center in each zone partitioned by those switching lines, where $f(x,y)=\sum _{i+j=0}^n{a}_{ij}{x}^i{y}^j$, $h(x)=\sum _{j=0}^l{b}_j{x}^j,a_{ij},b_j\in \mathbb{R},n,l\in \mathbb{N}$, and $g_m(x,y)$ 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 $L_m(n,l)$ for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system.