Turning Points and Applications

转折点和应用

• 批准号: 0406998
• 负责人: Weishi Liu
• 金额: $ 12.98万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406998&HistoricalAwards=false
• 关键词: Turning; Points; Applications

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.

项目申请：DMS-0406998PI：刘伟时；机构：堪萨斯大学题目：拐点与应用摘要本课题研究具有拐点的奇异摄动微分方程系统及其应用。目的是为这类系统建立一个综合的几何奇异摄动理论。奇摄动问题通常涉及导致非均匀行为的多尺度特征。拐点的存在导致稳定性的丧失，并造成动力结构的复杂性。研究者将使用动态系统方法系统地调查转折点行为。其动力学行为在很大程度上取决于拐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，并应用现代动力系统理论的分析和几何工具进行研究。多重时间和空间尺度使现实世界的系统丰富而令人兴奋，理解现实生活中的问题是数学理论发展的动力。该项目侧重于研究许多科学和工程领域产生的问题，如流体动力学、种群动力学、神经网络和生化过程，涉及多个时间和空间尺度。理论研究将确定对这种系统的复杂结构负责的关键参数，并检查过程中相互作用的数学形式。作为本提案的重要组成部分，将研究两个特定领域的应用：(1)生物系统中的松弛振荡，如捕食者-猎物模型和流行病模型；(2)废水处理的工程生化过程。这些系统在空间和时间上都涉及到非常不同的尺度，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。由于方法和结果的直观表述，拟议的活动也将大大加强对学生和非专家在这一重要领域的教育和培训计划。