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.

在本文中，我们研究了平面分段线性微分系统的极限环，其中两个区域被一条直线隔开。众所周知，当这些系统是连续的时，它们最多可以显示一个极限循环，而当它们不连续时，关于它们可以显示的最大极限循环数的问题仍然是未知的。对于这些最后的系统，有显示三个极限循环的示例。

本文的目的是研究一种特殊的平面不连续分段线性微分系统的极限环数，该系统具有两个被直线隔开的区域，被称为折射系统。首先，我们获得了焦点节点型折射系统极限环的存在性和唯一性。其次，我们证明焦点-焦点类型的折射系统最多具有一个极限环，因此我们对Freire等人关于Freire，Ponce和Torres提出的极限环唯一性的猜想给出了肯定的答案。（2013）。这两个结果完善了任何折射系统最多具有一个极限循环的证明。