Mathematical Sciences: Non-Autonomous Ordinary and Partial Differential Equations

非线性非自治微分方程引起了许多数学家和数学科学家的注意。物理现象的描述和建模经常涉及到这样的方程。取得进展并不容易，但拓扑和变分方法已经相对成功地获得了关于边值问题和周期解的存在性和多重性信息。然而，在我们对这类系统动力学的理解方面，进展要慢得多。这些问题的许多类别都有相当稀疏的文献，可能是由于拓扑度和变分方法通常很少或根本没有关于动力学的信息。本研究将通过同伦指标理论和分析方法，以及拓扑度、半群方法和单调迭代技术，对与非自治微分方程相关的参数依赖的偏积流，提出统一的解决这些困难的方法。将发展渐近稳定性、有界解的存在性和其他动力学问题的延拓方法。

Nonlinear nonautonomous differential equations have attracted the attention of many mathematicians and mathematical scientists. The description and modeling of physical phenomena frequently involve such equations. Progress has not been easy but topological and variational methods have been relatively successful in yielding existence and multiplicity information regarding boundary value problems and periodic solutions. However, progress has been much slower in our understanding of the dynamics of such systems. Many classes of these problems have a rather sparse literature, perhaps due to the fact that the topological degree and variational methods usually yield little or no information on dynamics. The proposed research will develop a unified approach to these difficulties through the application of homotopy index theory and analytical methods to parameter dependent skew product flows associated with the nonautonomous differential equations,as well as topological degree, semigroup methods, and monotone iterative techniques. Continuation methods for asymptotic stability, existence of bounded solutions and other dynamical questions will be developed.