Dynamical Systems, Stochastic Approximation and Applications

9802182 hirsch本项目将研究具有强比较原理的动力系统及其在进化随机近似过程中的应用。研究不变子集的结构，特别是吸引子和无吸引子集。给出了确定常微分方程和偏微分方程单调系统周期和次谐波轨迹的存在性、稳定性和周期的新方法。这些结果将应用于随机过程，其中样本路径的长期行为与相关确定性向量场的渐近行为密切相关，该确定性向量场指向状态向量的预期变化方向。在某些情况下，矢量场的动力学非常简单，当某些参数（例如，流行病模型中的人口）变大时，可以获得关于样本路径聚类的精确信息。具体应用包括传染病传播模型、价格调整模型和重复适应性博弈模型。对于许多现实世界的现象来说，拥有精确的数学模型来捕捉一些重要的特征是很有用的。例如，为了控制传染病的传播，重要的是要知道预算用于提高治愈率、降低接触率还是隔离受感染最严重的亚群。数学模型被用来对这些问题给出精确的理论答案；如果模型已被观察或实验验证，则可以采取具体的实际步骤。这个项目将在理论层面上研究这种类型的数学系统。技术结果也适用于其他情况。其中之一是价格调整和市场稳定；这将表明，当商品相似，价格变化很小，卖家对自己产品的内在边际需求有很好的估计时，价格可以预期稳定。另一种是信息不确定的重复游戏；这类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会导致行为趋同。

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.