可积、定性和分支理论在动力系统的动力学研究中起了重要的作用，本项目拟研究这些理论中几个重要的基础问题： 动力系统的局部和代数可积理论，以及这些理论在动力系统的动力学分析中的应用。这些研究内容部分是项目申请人前期研究工作的深入和发展，部分是新的研究课题，其中一些是长期没有解决的公开问题。具体研究内容如下：局部可积理论方面侧重于局部Darboux可积、以及解析微分系统在奇点和周期轨道邻域局部偏可积时系统的解析等价正规化子的存在性。整体可积理论方面侧重于Darboux可积理论的改进和发展，以及该理论与初等和刘维尔可积之间的本质联系；同时将这些理论应用于解决源于流体力学中的Kirchoff方程和Euler方程等实际模型的动力学。本项目还将研究Quaternion域上微分方程的代数可积性、及其局部和整体动力学。

The integrability, qualitative and bifurcation theories have played important roles in the dynamical analysis of dynamical systems, in this proposal we will study some basic important problems in these theories： local and algebraic integrabilities of dynamical systems，and their applications of these theories to dynamical analysis of dynamical systems. Parts of the project are the more deep development of the applicant’s previous research works and the other parts are completely new, which contain some long standing open problems, which cannot be solved until now. The concrete problems to be studied in this proposal are the following: on the local integrability we focus on the local Darboux theory of integrability of differential systems around a singularity, and the characterization on the existence of analytic normalization of locally analytic partial integrable differential systems in a neighborhood of either their singularity or their periodic orbit. On the global integrablity we mainly consider its further development of the Darboux theory of integrability, and the deep connection of this theory with the elementary and Liouvillian integrabilities; then we will apply these theories to analyze dynamics of the Kirchoff equation and Euler equation, which are originated from fluid mechanics. In this proposal we will also investigate algebraic integrability and their local and global dynamics of quaternion differential equations.