# Limit cycles appearing from the perturbation of differential systems with multiple switching curves

## TL;DR

This paper investigates how limit cycles bifurcate in a piecewise near-Hamilton system with multiple algebraic switching curves, providing bounds and conditions for their emergence under polynomial perturbations.

## Contribution

It offers new bounds on the number of bifurcating limit cycles in systems with multiple switching curves and analyzes the influence of switching curve configurations.

## Key findings

- At least 4 limit cycles can bifurcate for certain switching curves.
- Number of switching curves influences the maximum number of limit cycles.
- Upper bounds are established based on polynomial perturbation degree.

## Abstract

This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves $y=\pm x^2$. By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under $n$-th degree polynomial perturbations. In the case $n=1$, we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are $y=\pm x^2$ (resp. $y=x^2$ or $y=-x^2$). The results also show that the number of switching curves affects the number of limit cycles.