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

Abstract

We study the number of limit cycles of the piecewise smooth differential system $$\dot{x}=y, \quad \dot{y}=-x-\varepsilon\bigl(f(x)y+\operatorname{sgn}(x)g(x,y)\bigr),$$ where $f(x)$ is a polynomial of degree $n\geq 1$, $g(x,y)$ is a bivariate polynomial of degree $m\geq 2$, and $\varepsilon$ is a sufficiently small parameter. Placing the discontinuity on the vertical axis $Σ=\{x=0\}$ 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 $x^iy^j$ with both $i$ and $j$ odd contribute. Together with the first-order averaging theorem for discontinuous systems, this gives the lower bound $H_{m,n}=\lfloor n/2\rfloor+\lfloor m/2\rfloor$ 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.