微分方程的不可积性与动力学行为

长期以来，人们普遍认为不可积微分方程一般呈现出各种各样的复杂现象，但上述观点始终没有得到多少严格的论证。近年来的一些研究成果表明：方程的不可积性与拓扑熵、Melnikov积分以及Galois群的动力学行为之间有着密切的联系。本项目中，我们将结合微分Galois方法、Melnikov方法以及拓扑动力系统理论，研究微分方程的Galois不可积性所蕴含的动力学行为。研究内容包括：1) 探讨一般微分方程的Galois不可积性与其拓扑熵之间的联系，进而研究不可积微分方程的混沌等复杂行为。2) 研究(线性)微分方程的Liouville不可积性与相应Galois群的动力学性质之间的关系。3) 研究Galois不可积性与Melnikov积分之间的关系。4)发展能适当刻画和反映系统复杂行为的奇性分析方法。

For a long time, it is generally considered that non-integrable differential equations often exhibit a variety of complex phenomena, but there have been little strict argument for this observation so far. In recent years, several important works implied that there should be some close relationships between the topological entropy, the Melnikov integral and the differential Galois group for non-integrable differential equations. In this project, we will use the differential Galois theory, Melnikov method and the theory of topological dynamical systems to study the complex dynamical behavior of the Galoisian non-integrable differential equations. We shall: 1) investigate the link between the Galoisian non-integrability and the topological entropy of general differential equations; 2) find the relaionships between Liouville non-integrability and dynamical peoperty related to corresponding differential Galois group of linear differential equations;3) investigate relationships between the non-integrability and Melnikov integrals; 4) try to develop a singular analysis method for systematic studying the non-integrability and chaos of general differential equations .