Sharp estimates for the number of limit cycles in discontinuous generalized Liénard equations

Abstract

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider $g$ as a polynomial of degree $m$. We conclude that for sufficiently small values of $|ε|$, the number $\left[\frac{n}{2}\right]+\left[\frac{m}{2}\right]+1$ serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$. Furthermore, we demonstrate that it is indeed possible to achieve such a number of limit cycles.