In this paper we consider periodic orbits of planar linear Filippov systems with a line of discontinuity. Unlike many publications researching only the maximum number of crossing periodic orbits, we investigate not only the number and configuration of sliding periodic orbits, but also the coexistence of sliding periodic orbits and crossing ones. Firstly, we prove that the number of sliding periodic orbits is at most 2, and give all possible configurations of one or two sliding periodic orbits. Secondly, we prove that two sliding periodic orbits coexist with at most one crossing periodic orbit, and one sliding periodic orbit can coexist with two crossing ones.

中文翻译：

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具有不连续线的线性Filippov系统的周期轨道

在本文中，我们考虑具有不连续线的平面线性Filippov系统的周期轨道。与许多只研究交叉周期轨道的最大数目的出版物不同，我们不仅研究滑动周期轨道的数目和结构，而且研究滑动周期轨道和交叉轨道的共存。首先，我们证明了滑动周期轨道的数量最多为2，并给出了一个或两个滑动周期轨道的所有可能配置。其次，证明两个滑动周期轨道与一个交叉周期轨道共存，一个滑动周期轨道可以与两个交叉周期轨道共存。