Mathematical Sciences: Nonlinear Oscillations in Functional Differential Equations

数学科学：函数微分方程中的非线性振荡

• 批准号: 9101718
• 负责人: Harlan Stech
• 金额: $ 8.45万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101718&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Oscillations; Functional

This project concerns the development of numerical methods designed to assist the computer-aided analysis of nonlinear oscillation problems. The investigator undertakes work on the computation of normal forms for certain infinite dimensional systems, and a study of the relationship between that and a Lyapunov-Schmidt based technique developed by the investigator for the analysis of Hopf bifurcation problems. The use of symbolic manipulation software is expected to play an important role in this local analysis. Simultaneously and complementarily, the investigator will undertake significant modifications and improvements of existing simulation and numerical curve tracking software that he has developed. The issues addressed and techniques proposed are selected to promote the analysis of problems typified by systems possessing "time-delay" effects. Such so-called functional differential equations are important in their own right, and provide a transition towards analyzing analogous questions in problems from fluid dynamics. Specific problems include models of chugging in liquid propellant fuel rockets (where the time delay corresponds to the time it takes to vaporize fuel), oscillations in coupled (Josephson Junction) semiconductors, and pattern formation in diffusive chemical reactions. Both the analytic and computational aspects of this research are motivated by the goal of making the mathematical analysis of certain dynamical systems more accessible to nonspecialists in the field.

这个项目涉及开发数值方法，旨在帮助计算机辅助分析非线性振动问题。研究人员致力于计算某些无限维系统的规范形，并研究其与基于Lyapunov-Schmidt的Hopf分支问题分析技术之间的关系。预计符号处理软件的使用将在这一局部分析中发挥重要作用。同时，调查员将对他开发的现有模拟和数值曲线跟踪软件进行重大修改和改进。所讨论的问题和提出的技术是为了促进对具有“时滞”效应的系统的典型问题的分析。这种所谓的泛函微分方程本身就很重要，并提供了一种向分析流体动力学问题中的类似问题的过渡。具体问题包括液体推进剂燃料火箭中的嗡嗡声模型(其中时间延迟对应于燃料蒸发所需的时间)、耦合(约瑟夫森结)半导体中的振荡以及扩散化学反应中的图案形成。这项研究的分析和计算方面的动机都是为了使某些动力系统的数学分析更容易为该领域的非专家所用。