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Mathematical Sciences: Multiparameter Bifurcations and Applications

数学科学：多参数分岔及其应用

基本信息

批准号：
9022621
负责人：
Carmen Chicone
金额：
$ 6万
依托单位：
University of Missouri-Columbia
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1991
资助国家：
美国
起止时间：
1991-06-15 至 1994-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9022621&HistoricalAwards=false
关键词：
Mathematical Sciences Multiparameter Bifurcations Applications

项目摘要

This project will research bifurcation of multiparameter families of nonlinear differential equations. The primary focus is the analysis of bifurcations to periodic solutions. In the autonomous case the bifurcation of limit cycles from centers and from one parameter families of periodic solutions is considered while in the autonomous case bifurcation of subharmonic solutions is studied. A major goal is to develop an analysis that goes beyond the usual first order methods by being able to treat the response to a nonlinear perturbation to all orders. This will allow for a complete understanding of the bifurcation diagram of the multiparameter system by detecting periodic solutions which are invisible to first order methods and by mixing a priori information on the minimum order to which the perturbation analysis must be carried so as to exhaust the possibility of finding additional perturbed periodic solutions. The methods employed include geometric analysis of nonlinear dynamics, perturbation theory, "Andronov-Melnikov" theory, and algebraic geometric ideal theory. There is also an important computational component especially integration of differential equations, computer graphics and computer algebra. Applications of the theory are found wherever physical systems are modeled by nonlinear ordinary differential equations depending on parameters and, in particular, when determining the possible response of a nonlinear system to external stimuli. A typical application is illustrated in the project by the analysis of a wind oscillation problem.

本课题研究多参数系统的分岔问题非线性微分方程组主要重点是周期解的分叉分析。在自治情况下的极限环从中心和分支从一个参数族的周期解被认为是而在自治情形下，研究了一个主要的目标是开发一个分析，超越了通常的一阶方法，对所有阶次的非线性扰动的响应。这将允许完整理解的分叉图多参数系统的周期解检测方法，是不可见的一阶方法和混合先验关于扰动的最小阶数的信息必须进行分析，以排除寻找额外的扰动周期解。的方法所采用的方法包括非线性动力学的几何分析，微扰理论，“Andronov-Melnikov”理论，和代数几何理想理论还有一个重要的计算特别是微分方程的积分，计算机图形学和计算机代数。该理论的应用可以在任何物理系统由非线性常微分方程建模根据参数，特别是当确定非线性系统对外界刺激的可能反应。一通过分析说明了该方法在工程中的典型应用风的振荡问题。