微分方程的可积性是一个古老而基本的问题，在非线性系统的定性分析中起着重要作用。迄今仍有许多未解决的公开问题。本项目研究非线性微分方程可积性的一些有趣而重要的问题。具体研究内容如下：1.利用微分Galois理论建立三维解析系统不可积性的代数判断准则，探讨其Galois不可积性与Silnikov混沌复杂性之间的内在联系；2.系统地研究可逆动力系统的可积性或不可积性，并应用于Kirchhoff方程等数学物理模型的亚纯可积性分类问题；3.在几何奇异摄动理论的框架下, 研究描述细胞离子通道的具空间效应的Poisson-Nernst-Planck方程快慢极限系统的可积结构，进一步分析空间效应对于各离子通量、反转电流等重要物理量的影响，尝试为“门控”现象提供数学理论解释。

The integrability of nonlinear differential equations is an old and basic problem, which has played an important role in the qualitative analysis of the dynamical systems. Until now, there have been many open questions about this. In this project, we intend to solve some interesting and basic important problems in this field. Specifically, we will study the following contents: 1. Develop some algebraic methods for determining the non-integrability of 3D analytical system by the differential Galois theory, and explore the deep relationship between its Galois non-integrability and Silnikov chaos. 2. Systematically study the integrability or non-integrability of reversible dynamical systems, as applications to provide the complete integrability classifications of some physical models such as the Kirchhoff equations. 3. Under the framework of the geometric singular perturbation theory, characterize the integrable structure of limit systems of the Poisson-Nernst-Planck equations with steric effects for ion channels, further analyze the effects of ion sizes on important quantities such as the ion flux and reversal current, and try to provide a theoretical interpretation to the gating phenomenon.