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.
