This research will attempt to understand the effect of small noisy perturbations on certain types of complicated dynamical systems. The systems under study exhibit oscillations and bifurcations. For asymptotically small noise, one can in general find a stochastic process (via stochastic averaging) which encapsulates the evolution in the quotient space defined by the periodic oscillations; at the bifurcations, gluing conditions need to be added. The goal of the proposed research is to look at certain complicated systems with the above features. The investigation is roughly divided into three components. One part of the investigation will focus on the interaction of discontinuities with the above ideas. Another will focus on higher-dimensional systems arising from stability studies of simpler systems (i.e., tangent flows). A third will be the effect of noise upon certain systems for which the deterministic dynamics are commonly said to be "stochastic". This research attempts to better understand the effects of small noise upon certain complicated types of oscillatory systems. The systems under study all are important in various physical models. The above-mentioned investigation into discontinuities is motivated by certain vibro-impact systems, electrical networks with switches, and hybrid systems. Stability studies are crucial for design and engineering analyses. And deterministic "stochasticity" is intrinsic in many models of the climate, oceanic flow, and other important phenomena. In reality, noise is present in all of these systems, and a better understanding of its effect is clearly necessary.
