On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

In this paper we consider the linear differential center (x˙,y˙)=(−y,x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y=0.y=0. Using the Bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity. We also study the displacement function for a class of discontinuous piecewise smooth differential system.