本项目主要研究几类分段光滑系统异宿环的稳定性以及异宿环的分支问题.这些问题是分段光滑系统定性理论中的热点问题、难点问题，具有挑战性. 具体说来，我们将研究如下几个问题：(1)我们将首先研究一类分段光滑系统异宿环的稳定性问题，通过计算Poincare 映射的近似表达式，给出使得异宿环稳定的一些充分条件. (2)本项目还将研究一类分段光滑系统异宿环的分支问题，通过得出转移映射的近似表达式，给出分支出极限环个数的计算公式. 我们将运用全局分支理论.通过探索有效的变换、研究有效的估计零点个数的方法来解决这些问题. 在这两部分研究工作中，我们还将结合具体的例子来应用得到的结论.

This project is mainly to study the stability and bifurcation of the heteroclinic loops of several classes of piecewise smooth systems. These problems are the hot issues and difficult problems qualitative theory of piecewise smooth systems, they are challenging . To be concrete, we shall study the following questions: (1) We shall first study the stability of the heteroclinic loops of a class of piecewise smooth systems. By calculating approximate expression of Poincare mapping, we give some sufficient conditions for the stablility of its heteroclinic loop.(2) The project will also study the bifurcations of heteroclinic loops of a class of piecewise smooth systems, we shall give the number of limit cycles by finding approximate expression of the transition mapping. In the study of bifurcation problems of heteroclinic loops , we shall use the global bifurcation theory. We shall solve these problems by exploring some useful transformations and investigating some effective methods to estimate the number of zeros. Moreover, in the two parts of the work, we will combine some examples to apply our conclusions.