Mathematical Sciences: Nonlinear Oscillations in Functional Differential Equations

这个项目涉及数值方法的发展，旨在协助非线性振荡问题的计算机辅助分析。研究者从事某些无限维系统的范式计算工作，并研究其与研究者为分析Hopf分岔问题而开发的基于Lyapunov-Schmidt的技术之间的关系。符号操作软件的使用有望在这一局部分析中发挥重要作用。同时，作为补充，研究者将对其开发的现有仿真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，并为分析流体动力学问题中的类似问题提供了一个过渡。具体问题包括液体推进剂燃料火箭的轰隆声模型（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）半导体的振荡，以及扩散化学反应中的模式形成。本研究的分析和计算两方面都是为了使某些动力系统的数学分析更容易被该领域的非专业人士所接受。

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.