The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles. The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.