微分系统的中心、极限环分支与全局结构问题

综合运用Poincaré-Lyapunov方法、Poincaré紧致化法、定性理论与多项式运算相关的符号计算研究几类Zq等变对称微分系统的多中心条件和全局结构问题，研究幂零微分系统的中心条件、极限环个数与分布问题，研究微分系统的中心和扰动出极限环等定性性质与其对应的非线性波方程动力学性质的联系。. 在以上工作的基础上进一步深入开展非光滑微分系统的定性研究，研究具有高次切换曲线的分片光滑线性系统的极限环问题和全局结构问题，研究具有非初等型平衡点的分片光滑微分系统的中心判定、Hopf分支和极限环个数问题，如分片光滑幂零微分系统关于这些问题在已有的文献中还未被研究过。利用发展的方法与理论，分析一些实际领域中分片光滑微分模型的动力学性质。 . 上述所有问题的研究将对微分方程理论及其应用具有重要意义。

By using the Poincaré-Lyapunov method, Poincaré compactification, qualitative theory and symbolic calculation related with polynomial operations, we will study the multiple centers and global dynamics in the several classes of Zq-equivariant differential systems. We will research the center, the number and distribution of limit cycles for the nilpotent systems. And we investigate the connection between the differential systems and the corresponding nonlinear wave equations when these differential systems have center and bifurcation of limit cycles.. On the basis of the above researches, we further investigate the qualitative theory of non-smooth differential systems. We study the cyclity problem and global dynamics for piecewise smooth linear systems with high-order switching curves. In particular, we study the center problem, Hopf bifurcation and the number of limit cycles in piecewise smooth systems with non-primary equilibrium points. For example, these problems in piecewise smooth nilpotent systems are hardly seen in published literature. Using the developed theory and methods, we analyze the dynamics for some piecewise smooth systems, which come from the actual fields.. The above problems will be of great significance to the theory of differential equations and their applications.