Mathematical Sciences: Nonlinear Oscillations in Functional Differential Equations

这个项目涉及数值方法的发展 旨在帮助计算机辅助分析非线性 振荡问题。 调查员负责 一类无穷维规范形的计算 系统，并研究这与 研究者开发的基于Lyapunov-Schmidt的技术 用于分析Hopf分岔问题。 使用 符号处理软件预计将发挥重要作用， 在这个地方的分析。 同时和互补地， 研究者将进行重大修改， 现有模拟和数值曲线跟踪的改进 他开发的软件。 所处理的问题和提出的技术选择， 促进分析具有以下特点的系统的问题： “延时”效应。 这种所谓的功能性差异 方程本身就很重要，并提供了一个 过渡到分析问题中的类似问题， 流体动力学 具体的问题包括模式的突突在 液体推进剂燃料火箭（其中时间延迟对应于 到蒸发燃料所需的时间），耦合的振荡 （约瑟夫森结）半导体，以及在 扩散化学反应。 分析和 这项研究的计算方面的动机是， 对某些动力系统进行数学分析 更容易被非专业人士所接受。

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.