Averaging Approach to the Limit Cycles of A Class of Discontinuous Piecewise Differential Systems

These last years an increasing interest appeared for studying the discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study of their limit cycles. In this paper we study the limit cycles of the discontinuous piecewise differential systems separated by one straight line and formed by two distinct cubic reversible isochronous centers, whose first integrals are neither polynomial nor rational. We prove that 9 is the number of limit cycles of this kind of discontinuous piecewise differential systems that can be obtained using the averaging theory up to seven order.