利用反射函数理论来讨论四阶线性微分系统的下三角反射函数的存在性, 并计算出在不同情况下具体的反射矩阵.同时, 利用反射矩阵来建 立周期微分系统的庞加莱映射, 进而该系统周期解的存在稳定性判定定理也相应 地建立起来.最后, 将以上结果推广应用到了非线性微分系统中.

In this paper, we use the reflective function theory developed by Mironenko to discuss the existence of the lower triangular reflective matrix of the four-dimensional linear differential system, and then compute the specific reflective matrix under some given conditions. Meanwhile, we make use of this reflective matrix to establish the poincar\'{e} map of the $2\omega$-periodic system, and the decision theorem of the existence and stability of this system's periodic solution is established as well. In addition, we spread and apply these results of linear differential system to the non-linear differential system.