Limit Cycles of Discontinuous Piecewise Differential Systems Formed by Linear and Cubic Centers via Averaging Theory

Finding the number of limit cycles, as described by Poincare (Memoire sur les coubes definies par une equation differentielle, Editions Jacques Gabay, Sceaux, 1993), is one of the main problems in the qualitative theory of real planar differential systems. In general, studying limit cycles is a very challenging problem that is frequently difficult to solve. In this paper, we are interested in finding an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of a given discontinuous piecewise differential system when it is perturbed inside a class of polynomial differential systems of the same degree, by using the averaging method up to third order. We prove that the discontinuous piecewise differential systems formed by a linear focus or center and a cubic weak focus or center separated by one straight line y = 0 can have at most 7 limit cycles.