Mathematical Sciences: Applications of Bifurcation from Resonance

首席研究员建议继续他最近对非线性微分方程的研究。这项工作的主要理论重点是分析多维多参数系统在共振附近的周期解的分叉以及相关现象的描述，特别是通过共振的现象。所采用的方法包括非线性动力学的几何分析和经典微扰理论的几个方面：‘Melnikov’理论，Lyapunov-Schmidt约化，椭圆函数和小参数展开。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。该理论将应用于由耦合的非线性微分方程组模拟的物理系统。该项目的主要应用是研究直流电机驱动的机械连杆机构的运动。提出了确定周期运动的存在性和稳定性、反馈控制对周期运动的镇定以及控制参数在共振附近缓慢变化的影响。当机械或电气部件连接在一起时(例如，当电机连接到机器上时)，重要的是确定耦合系统的后续运行能力。在耦合系统的设计和运行控制中，可能会出现以下现象：其中一个部件产生的微小影响可能会对耦合系统的运行产生很大影响。这种现象可以用在电子设备中，例如放大器，但当它出现在机械中时，往往会产生严重的不良后果。连接到弹性支承上的电机可以平稳地运行，直到其转速的微小变化引起电机或其连杆的大振动和随后的故障。这类似于歌剧演员用声音打碎酒杯的经典例子。拟议的研究试图通过开发技术来分析这种耦合系统的运行的数学模型，从而找到确定机械或电气部件之间的耦合影响的方法。这些方法将在机械设计以及控制其运行所需的辅助系统的设计中有用。

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.