Dynamical Systems, Stochastic Approximation and Applications

9802182 Hirsch该项目将研究享受强比较原理的动力系统，以及它们在进化随机逼近过程中的应用。不变子集的结构将被研究，特别是吸引子和吸引子自由集。 将得到确定单调常微分方程和偏微分方程系统的周期轨线和次谐波轨线的存在性、稳定性和周期的新方法。 这些结果将被应用到随机过程中的长期行为的样本路径是密切相关的渐近行为相关联的确定性向量场，在预期的状态向量的变化的方向点。 在某些情况下，向量场的动力学足够简单，使得可以导出关于样本路径的聚类的精确信息作为某些参数（例如，流行病模型中的人口）变得很大。 具体的应用包括传染病的传播模型、价格调整和重复的自适应博弈。对于许多真实的世界现象，具有捕捉一些重要特征的精确数学模型是有用的。 例如，为了控制传染病的传播，重要的是要知道预算是用于提高治愈率、降低接触率还是用于隔离感染最严重的亚群。 数学模型被用来对这些问题给出精确的理论答案;如果模型已经被观察或实验所证实，就可以采取具体的实际步骤。本项目将在理论层面上研究这类系统。 技术成果也适用于其他情况。 其中之一是价格调整和市场稳定;它将表明，当商品相似、价格变化小、销售者对其产品的内在边际需求有很好的估计时，价格可望稳定下来。另一种是重复博弈，其中信息是不确定的;这种系统通常用于模拟个人，公司或政府之间的讨价还价。 该项目将表明，某些类型的长期战略导致行为的趋同。

9802182HirschThis project will study dynamical systems enjoying a strong comparison principle, and their applications to evolutionary stochastic approximation processes. The structure of invariant subsets will be investigated, especially attractors and attractor-free sets. New means of ascertaining the existence, stability and periods of periodic and subharmonic trajectories of monotone systems of ordinary and partial differential equations will be obtained. These results will be applied to stochastic processes in which long-term behavior of sample paths is closely related to asymptotic behavior of an associated deterministic vector field that points in the direction of expected changes in state vectors. In some cases the dynamics of the vector field is simple enough that one can derived precise information on clustering of sample paths as certain parameters (e.g., population, in epidemic models) becomes large. Specific applications include models of spread of infection disease, price adjustment, and repeated adaptive games.For many real world phenomena it is useful to have precise mathematical models capturing a few important features. In order to manage the spread of an infectious disease, for example, it is important to know whether it is better to budget for increasing cure rates, reducing contact rates, or isolating the most infected subgroups. Mathematical models are used to give precise theoretical answers to such questions; if the model has been validated by observation or experiment, specific practical steps can be taken. This project will investigate, on a theoretical level, mathematicalsystems of this type. The technical results apply also to other situations. One of these is price adjustment and stability of markets; it will be shown that prices can be expected to stabilize when goods are similar, price changes are small and sellers have good estimates of the intrinsic marginal demand for their own products. Another is to repeated games where information is uncertain; such systems are often used to model bargaining between individuals, corporations or governments. The project will show that certain types of long-term strategies lead to convergence of behavior.