Mathematical Sciences: Multiparameter Bifurcations and Applications

本项目将研究多参数非线性微分方程族的分支。主要的焦点是周期解的分叉分析。在自治情况下，研究了从中心到单参数周期解族的极限环分支，而在自治情况下，研究了次调和解的分支。一个主要的目标是开发一种超越通常的一阶方法的分析方法，能够处理对所有阶的非线性扰动的响应。这将允许通过检测一阶方法不可见的周期解和混合关于必须进行扰动分析的最小阶的先验信息来完全理解多参数系统的分岔图，从而用尽寻找附加扰动周期解的可能性。所采用的方法包括非线性动力学的几何分析、摄动理论、Andronov-Melnikov理论和代数几何理想理论。还有一个重要的计算部分，特别是微分方程式、计算机图形学和计算机代数的集成。在用依赖于参数的非线性常微分方程来模拟物理系统时，特别是在确定非线性系统对外部刺激的可能反应时，该理论的应用随处可见。通过对一个风振问题的分析，说明了该方法在工程中的典型应用。

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.