Mathematical Sciences: Topics in Theory and Application of Hopf Bifurcation

这是一个旨在开发用于分析Hopf分岔的科学软件的计算项目。在Hopf分岔理论中提出了三个不同领域的工作。1. 时滞微分方程Hopf分岔的定位与分析，以及这些方程的一般特征值问题。2. 时滞微分系统周期解延拓的数值算法。3. 常微分方程Hopf分岔的展开。第一个项目包括测试延迟微分方程系统线性稳定性的新方法，并将建立在现有的常微分方程软件的基础上。将该软件应用于时滞微分方程存在许多困难，其中之一是特征值的数值计算。第二个项目的目的是发展计算方法，以寻找与方程中参数的不同值相对应的周期解族。采用高斯-勒让德点配置的分段多项式近似和LU分解的一种变体等方法来解决这一问题。第三个项目涉及计算胚芽函数的高阶展开项，可能结合符号和数值计算。所有三个项目都有一些分析组成部分，并应用于各种其他研究领域中出现的特定方程。研究中包括的一个模型是霍奇金-赫胥黎电流固定神经传导模型。这些项目的目的是产生比所研究的具体系统更广泛适用的计算机代码。本研究属于常微分方程和泛函微分方程所描述问题的数学分析计算方法的一般领域。这类方程常用于描述物理、工程和生物学中的动力学现象。

This is a computational project aimed at developing scientific software for analysis of Hopf bifurcation. Work in three distinct areas within Hopf bifurcation theory is proposed. 1. Location and analysis of Hopf bifurcations in delay differential equations, and the general eigenvalue problem for these equations. 2. Numerical algorithms for continuation of periodic solutions of delay differential systems. 3. Unfolding of Hopf bifurcations for ordinary differential equations. The first project includes new methods for testing linear stability of systems of delay differential equations and will build on the existing software for the ordinary differential equations. Adaptation of this software to the case of delay differential equations presents a number of difficulties, one of them being the numerical computation of eigenvalues. The second project aims at developing computational methods for finding families of periodic solutions corresponding to various values of the parameter in the equation. Methods such as piecewise polynomial approximation with collocation at Gauss- Legendre points and a variant of LU decomposition will be used to solve the problem. The third project involves calculating higher order expansion terms of a germ fuction by possibly combining symbolic and numerical computing. All three projects have some analytic components and applications to specific equations arising in various other areas of research. One of the models included in the investigation is the Hodgkin-Huxley current-clamped nerve conduction model. These projects are intended to yield computer codes of wider applicability than the specific systems studied. This research falls into the general area of computational methods for mathematical analysis of problems described by ordinary and functional differential equations. Such equations are often used to describe dynamical phenomena in physics, engineering and biology.