# Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton system under perturbations of piecewise smooth polynomials

## TL;DR

This paper investigates how small perturbations in a specific class of integrable planar systems can lead to the emergence of up to six limit cycles, using averaging methods to establish bounds.

## Contribution

It provides a new upper bound of six limit cycles bifurcating from the global center under piecewise polynomial perturbations.

## Key findings

- Maximum of 6 limit cycles bifurcating from the period annulus.
- Use of average function of first order to analyze bifurcations.
- Establishment of sharper bounds for limit cycle bifurcations.

## Abstract

In this paper, we perturb the global center of the planar polynomial vector fields $\mathcal{X}(x,y)=(-y(x^2+a^2),x(x^2+a^2))$ ($a\neq0$) inside cubic piecewise smooth polynomials with switching line $y=0$. By using average function of first order, we prove that the sharper bound of the number of limit cycles bifurcating from the period annulus is 6.