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

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.