Random Perturbations of Complex Dynamical Systems

0305925 Sowers本研究将试图了解小噪声扰动对某些类型的复杂动力系统的影响。所研究的系统表现出振荡和分叉。对于渐近小的噪声，一般可以找到一个随机过程（通过随机平均），它封装了由周期振荡定义的商空间中的演化;在分叉处，需要添加胶合条件。所提出的研究的目标是看看某些复杂的系统与上述功能。调查大致分为三个部分。调查的一部分将集中在不连续性与上述想法的相互作用。另一个将侧重于从简单系统的稳定性研究中产生的高维系统（即，切向流）。第三个是噪声对某些系统的影响，对于这些系统，确定性动力学通常被称为“随机”。 本研究试图更好地理解小噪声对某些复杂类型的振荡系统的影响。所研究的系统在各种物理模型中都很重要。上述对不连续性的研究是由某些振动碰撞系统、带开关的电网络和混合系统引起的。稳定性研究对于设计和工程分析至关重要。确定性的“随机性”是许多气候、洋流和其他重要现象的模型所固有的。实际上，所有这些系统中都存在噪声，因此很有必要更好地了解其影响。

0305925Sowers 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.