Mathematical Sciences: Applications of Nonlinear Dynamics: Graviatational Ionization, Coupled Oscillators, Dielectric Response of Water, Dynamos

9303767奇科内首席研究员建议继续他最近的研究非线性微分方程。本文主要研究多维多参数系统在共振附近周期解的分岔现象，并对相关现象，特别是共振通道现象进行描述。 所采用的方法包括非线性动力学几何分析和经典扰动理论的几个方面：“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组件;特别是微分方程，计算机图形学和计算机代数的集成。 该理论将被应用到耦合的非线性微分方程系统建模的物理系统。本计画的主要应用是研究直流马达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，周期运动的反馈控制的稳定性和控制参数的慢变化的影响附近的共振。 当机械或电气部件连接在一起时（例如，当电机连接到机器时），重要的是确定耦合系统的后续操作能力。在耦合系统的设计和运行控制中，可能会出现以下现象：其中一个组件产生的小影响可能对耦合系统的运行产生很大影响。 这种现象可以在电气设备（例如放大器）中发挥作用，但当存在于机械中时，它通常会产生严重的不良后果。附接到弹性支撑件的马达可以平稳地操作，直到其旋转速度的小变化引起马达或其连杆的大振动和随后的故障。 这类似于一个歌剧演员用她的声音打破酒杯的经典例子。 拟议中的研究旨在通过开发用于确定机械或电气组件之间耦合影响的方法， 技术来分析这种耦合系统的操作的数学模型。 这些方法在机械设计以及控制其操作所需的辅助系统的设计中将是有用的。 ***

9303767 Chicone 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 a n 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. ***