Mathematical Sciences: Dynamical Systems and Applications

赫希 研究人员研究了动力系统及其应用的几个领域：（1）某些随机过程，如广义波利亚瓮现在已知有样本路径，其极限集几乎肯定是相关微分方程的链递归集。 这使得动力系统理论可以在各个领域中使用。 应用程序是随机逼近，博弈论，数学经济学的可持续资源，数值算法和神经学习算法。 (2)将证明马蹄混沌存在性的新方法应用于广义Henon映射。 推广Smal-Birkhoff存在定理，用一个更一般的非零相交数拓扑条件代替稳定流形和不稳定流形之间的横交假设. (3)基于振荡或混沌吸引子之间的不同竞争，构建了新的神经计算模型，竞争由自适应时钟系统控制。 数学动力系统理论研究复杂系统的长期行为。 该项目开发了严格的数学，可以帮助在几个应用程序中的系统分析。 (1)在经济理论中，市场通常被建模为竞争对手之间反复进行的游戏，这些竞争对手不断调整他们的策略（例如，价格，生产）。 确定这是否会导致稳定、可预测的局势至关重要。 该项目研究如何将这类问题转化为看似不同但很好理解的动力系统理论领域的数学问题来解决。 这个应用程序直接建立在最近的诺贝尔经济学奖获得者纳什和海萨尼的工作之上。 研究人员还与经济学家合作，将同样的数学结果应用于可持续资源模型。 (2)在科学工程的许多领域，重要的是特定的系统是否是“混沌的”，即从长远来看本质上是不可预测的。 这为确定混沌动力学提供了新的手段。 (3)新兴的神经计算领域在许多方面涉及动力学。 在本项目的其他部分中使用的一些相同的数学思想直接相关。 在与生物药理学家的合作中，研究人员将动力系统理论应用于新的神经网络系统，这些系统可以有效地“学习”执行各种有用的任务，例如机器人控制。

Hirsch The investigator studies several areas of dynamical systems and its applications: (1) Certain stochastic processes such as generalized Polya Urns are now known to have sample paths whose limit sets are almost surely Chain Recurrent sets for an associated differential equation. This allows the use of dynamical systems theory in a variety of fields. Applications are made to Stochastic Approximations, Game Theory, Mathematical Economics of sustainable resources, Numerical Algorithms, and Neural Learning Algorithms. (2) New methods for proving existence of Horseshoe Chaos are applied to generalized Henon maps. The Smale-Birkhoff existence theorem is extended to replace the assumption of transverse intersections between stable and unstable manifolds by a much more general topological condition of nonzero intersection number. (3) New models of neural computation are constructed, based on varying competition between oscillatory or chaotic attractors, with competition controlled by an adaptive clocking system. Mathematical dynamical systems theory studies the long-term behavior of complex systems. This project develops rigorous mathematics that can aid in the analysis of systems in several applications. (1) In Economic Theory, markets are often modeled as a game played repeatedly between competitors who continually adapt their strategies (e.g., prices, production) in the light of experience. It is of great importance to determine whether this leads to a stable, predictable situation. This project studies how questions of this kind can be solved by translating them into mathematical problems in the seemingly different but well understood field of dynamical systems theory. This application builds directly on work by the recent Nobel Laureates in Economics, Nash and Harsanyi. The investigator is also collaborating with economists on applications of the same mathematical results to models of sustainable resources. (2) In many areas of science en gineering it is important to whether particular systems are "chaotic," i.e. inherently unpredictable in the long run. This project provides new means for determining chaotic dynamics. (3) The newly emerging field of Neural Computation involves dynamics in many ways. Some of the same mathematical ideas used in the other parts of this project are directly relevant. In collaboration with biophysicists, the investigator applies dynamical systems theory to new neural network systems that can efficiently "learn" to do a variety of useful tasks, such as robot control.