On the number of limit cycles bifurcating from the linear center with an algebraic switching curve

This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^m$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an upper bound and an lower bound of the maximum number of limit cycles which bifurcate from the period annulus around the origin under polynomial perturbations of degree $n$. The results shows that the degree of switching curves affect the number of limit cycles.