This paper studies the bifurcations of limit cycles from the system \({\dot{x}}=y\) , \({\dot{y}}=-x\) with the switching curve \(y = x^3/3-x\) under the perturbations of arbitrary polynomials of x and y with degree n. We obtain the lower bound and upper bound of the maximum number of limit cycles bifurcating from \(h\in (0,3/2)\) if the first order Melnikov function is not identically 0. When the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first order Melnikov function. We also give an example to illustrate our result.
