非线性动力系统的Galois方法

近年来，随着Picard-Vissiot理论、Morales-Ramis理论和一般非线性系统的Galois可积理论的建立与完善，Galois方法在微分方程领域研究中显示出越来越重要的作用。因此，进一步拓宽Galois方法的应用范围，研究更一般的微分系统，或直接应用于研究非线性系统的动力学行为，具有重要的理论价值和应用意义。本项目将主要研究以下内容：1.运用Galois思想研究无穷维线性偏微分方程的积分可解性，探讨一般非线性发展方程的可积性问题；2.研究Lorenz系统和Rikitake系统等数学物理模型，尝试给出它们的可积性完全分类；探讨可逆系统的可积性与Darboux多项式、Jacobi乘子等特殊量之间的关系,尝试给出可积性新的判别准则。3.运用微分Galois理论建立一般非线性系统可积性与正则运动、不可积性与复杂动力学行为之间的具体联系，发展一套系统地研究动力学复杂性的代数方法。

Recent years, with the foundation and completion of the Picard-Vissiot theory, the Morales-Ramis theory and the Galoisian integrability theory for general dynamical systems, the Galoisian approach has shown an important effect in the study of differential equations. Therefore, it is a significant issue to extend the Galoisian approach to more general differential systems, or to analyze the dynamical behavior by the Galoisian approach directly. In this project, we will study the following problem: 1. Try to study the integrability of infinite dimensional evolution equations by using the Galoisian approach. 2. Give the classification of integrability for some mathematical physics models, such as Lorenz system, Nosé-Hoover equations, Rikitake systems and so on. Meanwhile, investigate the relationships between the integrability, Darboux polynomial and Jacobi multipliers for reversible system. 3. By using the differential Galoisian theory, investigate the algebric relationship between the integraility and dynamical behaviors, and try to obtain some new algebric method for studying the dynamical behaviors.