Mathematical Sciences: Applications of Bifurcation from Resonance

首席研究员建议继续他最近的 研究非线性微分方程。主 这项工作的理论重点是分析分叉， 多维非线性系统的周期共振解 多参数系统和相关现象的描述， 尤其是共振通道现象。 的 所采用的方法包括几何的几个方面， 非线性动力学和经典扰动理论的分析： "Melnikov“理论，Lyapunov-Schmidt还原， 椭圆函数和小参数展开。 有 也是一个重要的计算组件;特别是， 微分方程、计算机图形学和 计算机代数 该理论将被应用于由系统建模的物理系统 耦合非线性微分方程。主 本项目的应用是研究 直流电动机驱动的机械联动装置。 提出要 确定周期运动的存在性和稳定性， 反馈控制的周期运动镇定和 共振附近控制参数的缓慢变化的影响。 当机械或电气部件连接在一起时 (for例如，当电机连接到机器时）， 重要的是确定后续的作战能力， 耦合系统。在设计和操作控制中， 耦合系统中会出现以下现象： 其中一个组件产生的效果可能会产生很大的影响 对耦合系统的操作。 这种现象可 用于电子设备，例如放大器， 但当它出现时，通常会产生戏剧性的不良后果， 在机械方面。附接到弹性支撑件的马达可以操作， 直到其旋转速度的微小变化导致 电机的大振动和随后的故障，或 其联系。 这类似于一个经典的例子， 一位歌剧演员用她的声音打破了酒杯。 拟议 研究试图找到确定影响的方法， 机械或电气部件之间的耦合， 开发技术来分析数学模型， 这样的系统耦合。 这些方法将是有用的 在机械设计以及辅助设计中， 系统需要控制其运行。

The principal investigator proposes to continue his recent research on nonlinear differential equations. The primary theoretical focus of this work is the analysis of bifurcations to periodic solutions near resonance for multidimensional multiparameter systems and the description of related phenomena, especially the phenomenon of passage through resonance. The methods employed include several aspects of the geometric analysis of nonlinear dynamics and classical perturbation theory: ``Melnikov'' theory, Lyapunov-Schmidt reduction, elliptic functions and expansion in a small parameter. There is also an important computational component; especially, integration of differential equations, computer graphics and computer algebra. The theory will be applied to physical systems modeled by systems of coupled nonlinear differential equations. The primary application proposed for this project is to study the motions of a DC motor driven mechanical linkage. It is proposed to determine the existence and stability of periodic motions, the stabilization of periodic motions by feedback control and the effects of slow variation of control parameters near resonance. When mechanical or electrical components are linked together (for example when a motor is connected to a machine) it is important to determine the subsequent operational capabilities of the coupled system. In the design and operational control of coupled systems the following phenomenon can occur: a small effect produced by one of the components can have a large effect on the operation of the coupled system. This phenomenon can be used to advantage in electrical devices, for example amplifiers, but often it has dramatic undesirable consequences when present in machinery. A motor attached to an elastic support may operate smoothly until a small change in its rotational speed causes a large vibration and subsequent failure of the motor or its linkage. This is analogous to the classic example of an opera singer breaking a wine glass with her voice. The proposed research seeks to find methods to determine the effects of couplings among mechanical or electrical components by developing techniques to analyze mathematical models for the operation of such coupled systems. These methods will be useful in the design of machinery as well as the design of auxiliary systems needed to control its operation.