现实生活中许多模型都可以用非线性系统来描述， 因此对非线性系统作定性分析及分支研究是具有重要的理论与实际意义。本项目将对几类非线性系统的定性理论和极限环分支问题进行研究。主要研究以下三个方面。.1）探究拟奇次与半拟奇次系统的中心及闭轨分支问题。导出当权重固定时获得所有含中心奇点拟奇次多项式系统的算法并对极限环分支问题进行探讨；.2）研究含不变曲线或高次奇点多项式系统的相图及极限环分支问题。画出满足条件未扰动系统在平面上的相图, 给出围绕原点闭轨族的变化范围；对扰动系统的闭轨分支、Hopf 分支进行探讨；.3）探究含多条切换曲线分片光滑系统（该切换曲线是多条平行直线或者是多条过原点射线）的极限环分支问题。当未扰动系统含过多条切换曲线闭轨族时，导出扰动系统首阶 Melnikov 函数的显式表达式。此表达式可用来研究闭轨分支、Hopf 分支、同宿分支、异宿分支。

Many models in real life can be described by nonlinear systems, and it is very important to study the qualitative analysis and limit cycle bifurcations on these systems. In this project, we will investigate the qualitative theory and limit cycle bifurcations problems of several nonlinear systems. Our main contents consist of the following three aspects..Firstly, we study the center problem and Poincaré bifurcation on quasi-homogeneous and semi-quasi-homogeneous systems. An algorithm can be established to obtain all possible explicit systems for a given weight degree. We focus on center problems for such systems and provide some necessary conditions for the existence of centers. By the first-order Melnikov function method, we give the Poincaré cyclicity..Secondly, we research the phase portrait and limit cycle bifurcation problem of systems consists of multiple circles or higher order singular points. We plot all possible needed phase portraits and obtain the range field of closed orbits; Further, by using the first-order Melnikov function method, we study the Poincaré bifurcation and Hopf bifurcation of the unperturbed systems, obtaining their maximal numbers..Finally, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with n parallel switch lines or several straight lines passing through the same point. Under the restriction that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of Poincaré, Hopf, Homoclinic and heteroclinic bifurcations.