Limit Cycles in a 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 $n\ge 1$, $g(x,y)$ is a bivariate polynomial of degree $m\ge 2$, and $\epsilon$ is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number $H_{m,n}=\lfloor n/2\rfloor +\lfloor m/2\rfloor$ is a lower bound for the number of limit cycles bifurcating from the linear center $\dot{x}=y,\dot{y}=-x$. The bound is sharp, and we provide an explicit example that attains $H_{m,n}$ limit cycles.

arXiv comment3 figures. This work is based on the master's thesis of the second author