论文标题
平面分段线性差分系统的限制循环的统一上限，两个区域被直线分开
Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line
论文作者
论文摘要
对于平面分段线性差异系统的最大限制循环的均匀上限的存在，两个区域被直线隔开，这是数百篇论文感兴趣的主题。自Lum-Chua的工作以来进行了30多年的调查后，是否存在这种统一的上限是否存在，这仍然是一个悬而未决的问题。在这里，我们通过确定存在自然数量$ l^*\ leq 8 $的存在来给出一个积极的答案，任何平面分段线性差异系统具有两个由直线隔开的区域都不超过$ l^*$限制周期。通过结合新开发的线性差分系统半图的新开发的整体表征与Khovanski \Uı理论的扩展，以研究平滑曲线和特定类型的向量场之间的交叉点数量。
The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number $L^*\leq 8$ for which any planar piecewise linear differential system with two zones separated by a straight line has no more than $L^*$ limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincaré half-maps for linear differential systems with an extension of Khovanski\uı's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.
