Mathematical Sciences: Topics in Theory and Application of Hopf Bifurcation

这是一个计算项目， 科学软件来分析霍普夫分叉。工作 三个不同的领域内的霍普夫分歧理论提出。 1.时滞系统中Hopf分支的定位与分析 微分方程的一般特征值问题， 这些方程式。 2.延拓的数值算法 时滞微分系统的周期解 3.展开 常微分方程的Hopf分支。 第一个项目包括测试线性的新方法 时滞微分方程系统稳定性 在现有的常微分方程软件的基础上， 方程该软件的适应延迟的情况下 微分方程提出了一些困难，其中之一 它们是特征值的数值计算。 第二个项目旨在发展计算方法 寻找周期解族， 方程中参数的各种值。方法如 分段多项式逼近与配置在高斯- 勒让德点和LU分解的变体将用于 解决问题 第三个项目涉及计算高阶展开 通过可能结合符号和 数值计算 这三个项目都有一些分析成分， 应用于其他领域中出现的特定方程 的研究。 调查中包括的模型之一是 Hodgkin-Huxley电流钳神经传导模型。 这些项目旨在产生更广泛的计算机代码， 比研究的具体系统的适用性。 本研究福尔斯属于一般的计算领域 问题的数学分析方法， 常微分方程和泛函微分方程了该类方程 经常被用来描述物理学中的动力学现象， 工程学和生物学。

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.