On the limit cycles for a class of discontinuous piecewise cubic polynomial differential systems

This paper presents new results on the bifurcation of medium and small limit cycles from the periodic orbits surrounding a cubic center or from the cubic center that have a rational first integral of degree 2 respectively, when they are perturbed inside the class of all discontinuous piecewise cubic polynomial differential systems with the straight line of discontinuity y = 0. We obtain that the maximum number of medium limit cycles that can bifurcate from the periodic orbits surrounding the cubic center is 9 using the first order averaging method, and the maximum number of small limit cycles that can appear in a Hopf bifurcation at the cubic center is 6 using the fifth order averaging method. Moreover, both of the numbers can be reached by analyzing the number of simple zeros of the obtained averaged functions. In some sense, our results generalize the results in [Appl. Math. Comput. 250(2015), 887–907], Theorems 1 and 2 to the piecewise systems class.