Sharp estimates for the number of limit cycles in discontinuous generalized Li\'enard equations

TL;DR

This paper establishes a lower bound on the maximum number of limit cycles in a class of discontinuous Lie9nard systems using averaging methods, generalizing previous results and demonstrating the attainability of this bound.

Contribution

It extends existing results by considering polynomial g of degree m and provides explicit lower bounds for the number of limit cycles in discontinuous systems.

Findings

Lower bound of rac{n}{2}b1rac{m}{2}b1 1 for the number of limit cycles.

The bound is achievable in the studied system.

Generalization of previous Lie9nard system results.

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 Li\'enard systems. In our generalization, we consider $g$ as a polynomial of degree $m$. We conclude that for sufficiently small values of $|\epsilon|$, 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.