Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Systems with Bivariate Perturbation

We study the number of limit cycles of the piecewise smooth differential system $$\dot{x}=y,\dot{y}=-x-\epsilon (f(x)y+\mathrm{sgn}(x)g(x,y)),$$ where $f(x)$ is a polynomial of degree $n\ge 1$, $g(x,y)$ is a bivariate polynomial of degree $m\ge 2$, and $\epsilon$ is a sufficiently small parameter. Placing the discontinuity on the vertical axis $\mathrm{\Sigma }=\{x=0\}$ forces a parity obstruction: if $g$ depends only on $x$, its contribution to the first-order averaged function vanishes identically, regardless of the degree of $g$. For bivariate $g(x,y)$, only monomials $x^i{y}^j$ with both $i$ and $j$ odd contribute. Together with the first-order averaging theorem for discontinuous systems, this gives the lower bound $H_{m,n}=\lfloor n\mathrm{/}2\rfloor +\lfloor m\mathrm{/}2\rfloor$ for the number of limit cycles bifurcating from the linear center. The bound is sharp, and we provide an explicit example that attains $H_{m,n}$ limit cycles.