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

数学科学：时滞参数相关系统周期解定位的局部和全局技术

• 批准号: 8701456
• 负责人: Harlan Stech
• 金额: $ 6万
• 依托单位: University of Minnesota Duluth
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701456&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Local; Global; Techniques

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.

该项目的目的是开发新的数值技术 用于研究非线性微分方程的周期解 具有时间延迟。这种方程的理论已经在 在过去的二十年里，应用于各种动力系统的研究， 工程、生物科学和经济学。这些方程 也称为泛函微分方程。其中一个现象是 实践的重要性和理论的兴趣是存在的， 振荡，在数学上被描述为周期解。 该调查员将开展三个次级项目。首先他 我将翻译先前已知的结果，所谓的霍普夫分歧 （由于小的变化而从非周期性到周期性的变化） （一个参数）转换成新的计算方法。研究者已 最近开发了一种技术来检测小周期性的存在， 解决方案，避免了在无限的方程分析 维状态空间他的方法是基于对标量的分析 “分叉函数”，其可以以非常直接的方式计算。 他现在建议将这种技术的使用扩展到更普遍的领域。 沃尔泰拉积分方程、泛函微分方程 中立型方程和抽象泛函微分 方程 其次，调查人员将开发算法来定位 数值计算的参数值的霍普夫分岔 发生。他将扩展现有的方法来计算这样的分歧 通过分析线性化系统的特征方程， 并且通过执行左手的范数的数值最小化 的方面想他还将使用曲线跟踪程序，类似于那些 为常微分方程而开发的。他还将开发 下的方程的Floquet乘子的计算方法 研究，采用搭配方法。这项工作的大部分在技术上是 困难并且需要大量的计算技能和资源。 调查人员将使用位于主校区的Cray-2超级计算机， 明尼阿波利斯的明尼苏达大学。 第三，调查员将研究一些具体的例子 方程中出现的应用，并将测试他的数值方法 在这些例子上。这项工作的结果预计将提供新的 对其他一些动力学现象的解释 学科 位于杜卢斯的明尼苏达大学主要是本科生 这是一个有着学生参与研究的良好传统的机构。 在这个项目中，学生将被雇用来协助编程， 计算。他们对这项研究的参与将产生强烈的兴趣。 对他们的教育产生积极的影响，特别是由于各种 根据申请对结果的可能解释 其他科学。 该项目将与美国空军办公室联合资助。 科研