# Limit cycles by perturbing quadratic isochronous centers inside piecewise smooth polynomial differential systems

## TL;DR

This paper investigates the number of limit cycles bifurcating from quadratic isochronous centers in piecewise smooth polynomial differential systems by analyzing the zeros of the Melnikov function, providing explicit bounds and improving existing estimates.

## Contribution

It offers explicit upper bounds for the zeros of the Melnikov function for quadratic isochronous centers and improves previous bounds for certain cases within polynomial differential systems.

## Key findings

- Provided explicit upper bounds for the number of zeros of Melnikov functions.
- Improved bounds from previous studies for quadratic isochronous center S4.
- Found evidence of the equivalence between Melnikov and Averaged functions in this context.

## Abstract

In the present paper, we study the number of zeros of the first order Melnikov function for piecewise smooth polynomial differential system, to estimate the number of limit cycles bifurcated from the period annulus of quadratic isochronous centers, when they are perturbed inside the class of all piecewise smooth polynomial differential systems of degree $n$ with the straight line of discontinuity $x=0$. An explicit and fairly accurate upper bound for the number of zeros of the first order Melnikov functions with respect to quadratic isochronous centers $S_1, S_2$ and $S_3$ is provided. For quadratic isochronous center $S_4$, we give a rough estimate for the number of zeros of the first order Melnikov function due to its complexity. Furthermore, we improve the upper bound associated with $S_4$, from $14n+11$ in \cite{LLLZ}, $12n-1$ in \cite{SZ} to $[(5n-5)/2]$, when it is perturbed inside all smooth polynomial differential systems of degree $n$. Besides, some evidence on the equivalence of the first order Melnikov function and the first order Averaged function for piecewise smooth polynomial differential systems is found.