Turning Points and Applications

提案：论文题目：转折点及其应用摘要本项目主要研究具有转折点的奇摄动微分方程组及其应用。我们的目标是建立一个全面的几何奇异摄动理论这样的系统。 奇异摄动问题通常涉及多尺度特征，导致非均匀行为。转折点的存在导致稳定性的丧失，并在动态结构中产生复杂性。 研究者将使用动态系统方法系统地研究转折点行为。 动力学行为很大程度上取决于转向点集的拓扑结构及其与向量场的关系。研究者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于其研究中。多个时间和空间尺度使现实世界的系统变得丰富和令人兴奋，而理解现实生活中的问题是数学理论发展的动力。该项目的重点是研究从许多科学和工程领域产生的问题-如流体动力学，人口动力学，神经网络和生物化学过程-并涉及多个时间和空间尺度。 理论研究将确定负责这种系统的复杂结构的关键参数，并研究过程中相互作用的数学形式。 作为这项建议的一个重要组成部分，将审查两个具体的应用领域： (1)生物系统中的弛豫振荡，如捕食者-被捕食者模型和流行病模型;（2）废水处理工程生化过程。 这些系统在空间和时间上涉及非常不同的尺度，并表现出复杂的行为。 该项目如果成功，将大大提高我们对多尺度物理现象的理解，并为更好的控制工程设计提供见解。 拟议的活动还将大大加强这一重要领域对学生和非专家的教育和培训方案，因为其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.