Limit Cycles Bifurcating From Discontinuous Polynomial Pertubations of Higher Dimensional Linear Differential Systems

We study the periodic solutions bifurcating from periodic orbits of linear differential systems x0 = Mx, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the differential system x0 = Mx + "F n 1 (x) + "2Fn2 (x);
in Rd+2 where " is a small parameter, M is a (d+2) x (d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d - m
non{zero real eigenvalues. For solving this problem we need to extend the averaging theory for studying periodic solutions to a new class of non{autonomous d + 1-dimensional discontinuous piecewise smooth differential system.