Turning Points and Applications

建议：dms-0406998PI：微视刘研究所：堪萨斯大学题目：转折点及其应用本项目研究具有转折点的奇异摄动微分方程组及其应用。目标是建立这类系统的一个全面的几何奇异摄动理论。奇异摄动问题通常涉及导致非均匀行为的多尺度特征。转折点的存在会导致稳定性的丧失，并在动态结构中造成复杂的情况。研究人员将使用动力系统方法系统地研究转折点行为。动力学行为在很大程度上取决于转折点集合的拓扑结构及其与矢量场的关系。研究人员将对主要结构进行分类，并将现代动力系统理论的分析和几何工具应用到它们的研究中。多时空尺度使现实世界的系统丰富和令人兴奋，理解现实生活中的问题是数学理论发展的动力。这个项目侧重于研究许多科学和工程领域出现的问题--如流体动力学、种群动力学、神经网络和生化过程--并涉及多个时间和空间尺度。理论研究将确定导致这类系统复杂结构的关键参数，并检查过程中相互作用的数学形式。作为该建议的一个重要组成部分，将考察两个具体的应用领域：(1)捕食者-食饵模型和流行病模型等生物系统中的松弛振荡；(2)污水处理的工程生化过程。这些系统涉及非常不同的尺度，无论是在空间上还是在时间上，并表现出复杂的行为。这个项目如果成功，将显著提高我们对多尺度物理现象的理解，并为更好地进行控制目的的工程设计提供洞察力。拟议的活动还将极大地加强针对学生和非专家的这一重要领域的教育和培训方案，因为方法和结果都是直观的。

Proposal: DMS-0406998PI: Weishi LiuInstitution: University of KansasTitle: Turning Points and Applications ABSTRACTThis project is concerned with singularly perturbed systems of differential equations with turning points and their applications. The goal is to establish a comprehensive geometric singular perturbation theory for such systems. Singularly perturbed problems typically involve multiple-scale features that result in non-uniform behavior. The presence of turning points causes a loss of stability and creates complications in the dynamical structure. The investigator will use a dynamical-systems approach to systematically investigate turning-point behavior. The dynamical behavior depends heavily on the topological structure of the set of turning points and its relation to the vector field. The investigator will classify the main structures and apply analytical and geometric tools of modern dynamical systems theory to their study.Multiple time and space scales make real-world systems rich and exciting, and understanding real-life problems is the driving force for the development of mathematical theory. This project focuses on the study of problems arising from many areas of science and engineering--such as fluid dynamics, population dynamics, neural networks, and biochemical processes--and involving multiple time and space scales. The theoretical study will identify critical parameters responsible for the complicated structure of such systems and examine mathematical forms of interactions in the processes. As an important component of this proposal, two specific fields of applications will be examined: (1) relaxation oscillations in biological systems such as predator-prey models and epidemic disease models, and (2) engineered biochemical process of wastewater treatment. These systems involve vastly different scales, both in space and in time, and demonstrate complicated behaviors. This project, if successful, will significantly improve our understanding of physical phenomena with multiple scales and provide insight for better engineering designs for control purposes. The proposed activity will also greatly enhance education and training programs in this important area for students and non-experts because of the intuitive formulation of the approach as well as the results.