On the Number of Limit cycles for Discontinuous piecewise Linear differential Systems in ℝ2n with Two Zones

For a given differential system, a limit cycle is a periodic orbit isolated in the set of all its periodic orbits. Within the qualitative theory of differential systems, the study of their limit cycles is one of the main topics. Many questions are considered on the limit cycles of the differential systems in R2. Thus, one of the main lines of research for such systems is the study of how many limit cycles emerge from the periodic orbits of a center when we perturb it inside a given class of differential equations, see for example, [Christopher & Li, 2007] and the references therein. More precisely, the problem is to consider the planar linear differential center ẋ = −y, ẏ = x and perturb it ẋ = −y + εf(x, y), ẏ = x+ εg(x, y),