动力系统量和特征值的强连续性及最优估计

In this project, we are concerned with the strong continuous dependence of macro quantities defined from differential equations on micro quantities, together with the extremal problems and optimal estimations of macro quantities. By micro quantities, it means potentials and densities in equations, and macro quantities include Lyapunov exponents and rotation numbers in dynamical systems, as well as eigenvalues, half-eigenvalues and Fucik curves in spectral theory. By strong continuity, it means that macro quantities are continuous in micro quantities when the weak topologies are considered for micro quantities. Such a strong continuity is much stronger than the continuity in the usual sense when the norm topologies are considered for micro quantities. Based on some important works we have done for ordinary differential equations, we will systematically develop strong continuity results. As an application, by exploiting the geometric and topological structures of infinite dimensional spaces, with the help of nonlinear analysis such as variational methods, we will solve several important, non-standard and difficult extremal problems concerning with these macro quantities, especially with dynamics quantities. These results can yield natural optimal estimations for these macro quantities. On one aspect the present work will sharpen our understanding on linear systems, and on the other aspect it will become the foundation of our study of dynamics and analysis problems for nonlinear equations.

本项目研究由微分方程所定义的宏观量与微观量之间的强连续依赖关系以及这些量的极值问题和最优估计。这里的微观量是指方程中的位势、密度等，而宏观量包括动力系统中的Lyapunov指数、旋转数等和谱理论中的特征值、半特征值和Fucik曲线等。所谓强连续性是指对弱拓扑下变化的微观量的连续性，这是强于正常拓扑下看微观量的连续性结果。基于前期关于常微分方程方面的工作，我们将系统地发掘这些量的强连续性结果。进一步地，利用无穷维空间的强弱拓扑下的几何和拓扑结构，结合包括变分方法在内的非线性分析方法，我们将解决涉及这些宏观量（尤其是动力系统量）的若干重要而又在分析上非常困难的极值问题。这些定量性的结果将自然地导出对于这些宏观量的最优估计。本项目的研究可以加深对于线性系统的理解，并为我们进一步研究非线性系统的动力学问题和相关的分析问题奠定良好的基础。

本项目研究了由微分方程所定义的宏观量（旋转数、特征值、半特征值和Fucik曲线等）与微观量（位势、密度等）之间的强连续依赖关系以及这些量的极值问题和最优估计。这里的强连续性是指对弱拓扑下变化的微观量的连续性，这是强于正常拓扑下看微观量的连续性结果。我们系统地发掘了这些宏观量的强连续性结果。进一步地，利用无穷维空间的强弱拓扑下的几何和拓扑结构，结合包括变分方法在内的非线性分析方法，我们解决了涉及这些宏观量（尤其是动力系统量）的若干重要而又在分析上非常困难的极值问题。定量性的结果自然地导出了对于这些宏观量的最优估计。本项目的研究加深了我们对于线性系统的理解，并为进一步研究非线性系统的动力学问题和相关的分析问题提供了新的思路和方法、奠定了良好的基础。