Mathematical Sciences: Dynamical Systems and Applications

Hirsch研究了动力系统的几个领域及其应用：(1)某些随机过程，如广义Polya骨灰盒，现在已知存在其极限集几乎必然是相伴微分方程的链递归集的样本路径。这使得动力系统理论可以在各种领域中使用。应用于随机逼近、博弈论、可持续资源的数学经济学、数值算法和神经学习算法。(2)将证明马蹄形混沌存在性的新方法应用于广义Henon映射。推广了Smer-Birkhoff存在定理，用非零交数这一更一般的拓扑条件代替了稳定流形和不稳定流形的横交假设。(3)基于振荡吸引子或混沌吸引子之间的变化竞争，构造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。这个项目开发了严谨的数学，可以在几个应用中帮助分析系统。(1)在经济学理论中，市场经常被建模为竞争者之间反复进行的博弈，竞争者根据经验不断调整自己的策略(例如，价格、生产)。确定这是否会导致稳定、可预测的局势是非常重要的。这个项目研究如何通过将这类问题转化为动力系统理论中看似不同但却很好理解的数学问题来解决。这个应用程序直接基于最近的诺贝尔经济学奖获得者纳什和哈萨尼的工作。这位研究人员还与经济学家合作，将相同的数学结果应用于可持续资源的模型。(2)在科学工程的许多领域，重要的是特定的系统是否“混乱”，即从长远来看，本质上是不可预测的。该项目为确定混沌动力学提供了新的手段。(3)神经计算这一新兴领域在很多方面都涉及到动力学。在这个项目的其他部分中使用的一些相同的数学思想是直接相关的。在与生物物理学家的合作下，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进行学习，以完成各种有用的任务，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.