Dynamical Systems: from Theory to Applications

动力系统：从理论到应用

• 批准号: 1101594
• 负责人: Lai-Sang Young
• 金额: $ 37.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101594&HistoricalAwards=false
• 关键词: Dynamical; Systems; Theory; Applications

Five research projects in dynamical systems theory are undertaken. The first three are focused on advances within the subject.The goal of Project 1 is to extend nonuniform hyperbolic theory (or the mathematical theory of chaos), from finite to infinite dimensions, enlarging its range of applicability to include certain classes of evolutionary partial differential equations. Project 2 is about chaotic behavior in randomly perturbed dynamical systems, and Project 3 seeks to extend certain theories of autonomous dynamical systems to time-dependent systems with slowly drifting coefficients. The last two projects export dynamical systems ideas to two disciplines outside of mathematics: nonequilibrium statistical mechanics and theoretical neuroscience. Project 4 addresses issues related to the analog of local temperature in dynamical models that are driven out of equilibrium. Project 5 pertains to neuronal networks, seeking dynamical explanations for various phenomena observed by experimentalists.The Principal Investigator works with systems that are large, random, or out of equilibrium, three characteristics shared by many real-world systems. Scientifically, the proposed research will (1) lead to significant advances in the theory of dynamical systems, (2) build connections between dynamical systems and other branches of mathematics, such as partial differential equations and probability, and (3) broaden the use mathematics, the applicability of dynamical systems ideas in particular, in engineering, theoretical physics and neuroscience. The cross-fertilization in (2) and (3) will be beneficial to all the areas involved. On the educational front, providing scientific training and guidance for young researchers is an important part of the proposed activity. The Principal Investigator expects to continue this activity using projects in the current proposal, and to continue her public speaking and writing of expository articles as part of the process for dissemination of ideas.

承担了动力系统理论的五个研究项目。项目1的目标是将非均匀双曲理论(或混沌的数学理论)从有限维扩展到无限维，将其适用范围扩大到包括某些类型的发展偏微分方程组。项目2是关于随机扰动动力系统中的混沌行为，项目3试图将自治动力系统的某些理论扩展到具有缓慢漂移系数的依赖时间的系统。最后两个项目将动力系统的思想输出到数学之外的两个学科：非平衡统计力学和理论神经科学。项目4解决了与被驱使失去平衡的动力学模型中的局部温度的模拟有关的问题。项目5涉及神经元网络，寻求对实验者观察到的各种现象的动力学解释。首席调查者研究的是大型、随机或不平衡的系统，这三个特征是许多现实世界系统所共有的。从科学上讲，拟议的研究将(1)导致动力系统理论的重大进展，(2)建立动力系统与偏微分方程和概率等其他数学分支之间的联系，以及(3)扩大数学的使用，特别是在工程、理论物理和神经科学中动力系统思想的适用性。(2)和(3)中的交叉受精对所有涉及的地区都是有利的。在教育方面，为青年研究人员提供科学培训和指导是拟议活动的重要组成部分。首席调查员希望利用当前提案中的项目继续开展这项活动，并继续公开演讲和撰写说明性文章，作为传播思想进程的一部分。