论文标题
平面分段线性差异系统中极限周期的独特性和稳定性而无需滑动区域
Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region
论文作者
论文摘要
在本文中，我们考虑了平面分段线性差异系统的家族，其两个区域被两个区域隔开，而无需滑动区域，即，除了最多一个点，其流动在横向横断开关线的差速器系统。在研究文献中，许多论文涉及确定这些差异系统可以拥有的最大限制周期数量的问题。通常，通过大型大小写分析来解决此问题，该分析区分了差异系统矩阵光谱的许多不同可能性。在这里，通过使用庞加莱半图的新型积分表征，我们证明，矩阵光谱的不必要区分，对于这些差异系统的极限周期数的最佳均匀上限是一个。此外，事实证明，如果存在该限制周期是双曲线，其稳定性是由系统参数的简单条件确定的。作为我们分析的副产品，还得出了限制周期存在的条件。
In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.
