On the Number of Limit Cycles Bifurcating from the Linear Center with a Cubic Switching Curve

This paper studies the bifurcations of limit cycles from the system x ˙ = y , 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 ∈ ( 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.