On the number of limit cycles for a class of discontinuous quadratic differential systems

TL;DR

This paper investigates the maximum number of limit cycles that can bifurcate from a quadratic isochronous center under discontinuous quadratic perturbations, establishing that this maximum is five and is achievable.

Contribution

It provides a definitive upper bound of five limit cycles for a class of discontinuous quadratic systems, using the Chebyshev criterion, and completes previous open questions.

Findings

Maximum of five limit cycles bifurcated from the center.

The maximum number five is realizable in practice.

The results fully answer previous open questions.

Abstract

The present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center $\dot{x}=-y+\frac{16}{3}x^{2}-\frac{4}{3}y^{2},\dot{y}=x+\frac{8}{3}xy$ by the averaging method of first order, when it is perturbed inside a class of discontinuous quadratic polynomial differential systems. The \emph{Chebyshev criterion} is used to show that this maximum number is 5 and can be realizable. The result and that in paper \cite{LC} completely answer the questions left in the paper \cite{LM}.