Mathematical Sciences: Local and Global Techniques for the Location of Periodic Solutions of Parameter Dependent Systems with Time Delay

本项目的目的是发展新的数值方法来研究非线性时滞微分方程周期解。这类方程的理论是近二十年来发展起来的，并应用于工程、生物科学和经济学中的各种动力系统的研究。这些方程也称为泛函微分方程。具有实际意义和理论意义的现象之一是振荡的存在，在数学上被描述为周期解。这位调查员将执行三个分项目。首先，他将把早先关于所谓Hopf分叉(由于参数的微小变化而从非周期解转变为周期解)的已知结果转化为新的计算方法。研究者最近开发了一种技术来检测小周期解的存在，避免了在无限维状态空间中分析方程。他的方法是基于对标量“分支函数”的分析，这种分析可以非常直接地计算。他现在建议将这一技术的使用扩展到更一般的方程，如Volterra积分方程、中立型泛函微分方程和抽象泛函微分方程。其次，研究人员将开发算法来数值定位发生Hopf分叉的参数值。他将扩展现有的方法，通过分析线性化系统的特征方程，并通过执行左侧范数的数值最小化来计算这种分叉点。他还将使用曲线跟踪程序，类似于为常微分方程式开发的程序。他还将利用配置法开发计算所研究方程的弗洛奎乘子的方法。这项工作中的大部分在技术上是困难的，需要大量的计算技能和资源。调查人员将在明尼阿波利斯明尼苏达大学主校区使用一台Cray-2超级计算机。第三，研究人员将研究一些在应用中出现的具体例子方程，并将在这些例子上测试他的数值方法。这项工作的结果有望为在许多其他学科中遇到的动力学现象提供新的解释。明尼苏达大学德卢斯分校是一所以本科生为主的机构，有着学生参与研究的良好传统。在这个项目中，学生将被雇用来协助编程和计算。他们参与这项研究将对他们的教育产生强烈的积极影响，特别是因为对结果的各种可能的解释是基于对其他科学的应用。该项目将与空军科学研究办公室联合资助。

The purpose of this project is to develop new numerical techniques for the study of periodic solutions of nonlinear differential equations with time delay. The theory of such equations has been developed in the last twenty years and applied to study of various dynamical systems in engineering, biological sciences, and economics. These equations are also called functional differential equations. One of the phenomena of practical importance and theoretical interest is the existence of oscillations, mathematically described as periodic solutions. Three sub-projects will be pursued by this investigator. First, he will translate earlier known results on the so called Hopf bifurcation (a change from non-periodic to periodic solutions due to a small change of a parameter) into new computational methods. The investigator has recently developed a technique to detect the existence of small periodic solutions which avoids the analysis of equations in the infinite dimensional state space. His method is based on the analysis of a scalar "bifurcation function", which can be computed in a very direct manner. He now proposes to extend the use of this technique to more general equations such as Volterra integral equations, functional differential equations of neutral type, and abstract functional differential equations. Second, the investigator will develop algorithms to locate numerically the values of parameters for which the Hopf bifurcation occurs. He will extend the existing methods to compute such bifurcation points by analyzing the characteristic equation of a linearized system, and by performing a numerical minimization of the norm of the left hand side. He will also use curve tracking procedures, similar to those developed for ordinary differential equations. He will also develop methods for computing the Floquet multipliers for the equations under study, by using collocation methods. Much of this work is technically difficult and requires substantial computational skill and resources. The investigator will use a Cray-2 supercomputer at the main campus of the University of Minnesota in Minneapolis. Third, the investigator will study a number of concrete examples equations arising in applications, and will test his numerical methods on these examples. The results of this work are expected to provide new interpretations for dynamical phenomena encountered in a number of other disciplines. The University of Minnesota at Duluth is a primarily undergraduate institution which has good traditions of student involvment in research. In this project students will be employed to assist with programming and computations. Their involvment in this research will have a strong positive impact on their education, especially because of various possible interpretations of the results on the grounds of applications to other sciences. The project will be funded jointly with the Air Force Office of Scientific Research.