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This paper studies the family of piecewise linear differential systems in theplane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ isan arbitrary positive. By analysing the first order Melnikov function, we givean upper bound and an lower bound of the maximum number of limit cycles whichbifurcate from the period annulus around the origin under polynomialperturbations of degree $n$. The results shows that the degree of switchingcurves affect the number of limit cycles.

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