CAREER: Integrating Geometric, Probabilistic, and Topological Methods for Phase Space Transport in Dynamical Systems

职业：集成几何、概率和拓扑方法用于动力系统中的相空间传输

基本信息

批准号：
1150456
负责人：
Shane Ross
金额：
$ 42万
依托单位：
Virginia Polytechnic Institute and State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-06-01 至 2018-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1150456&HistoricalAwards=false
关键词：
CAREER Integrating Geometric Probabilistic Topological

项目摘要

The research objective of this Faculty Early Career Development (CAREER) project is to develop a firm unified foundation for the theoretical and computational analysis of the phase space transport problem. Many physical systems can be modeled as having an underlying dynamical skeleton which organizes how all the possible behaviors are related. For systems known analytically with simple time dependence, this skeleton consists of well-known mathematical structures such as invariant sets and sets which asymptotically approach or depart from such sets. But for systems of increasing interest which are non-periodic, based on data, and observed over only a short time, there are no such structures in general. The proposed research will find appropriate analogs to these structures to expand the applicability of dynamical systems results to real world data. Working in the context of fluid flows, we will develop tools using geometric, probabilistic, and topological approaches, including coherent sets, almost-invariant sets, and symbolic dynamics.Successful completion of this project will provide a new and fruitful approach for conceptualization, visualization, and extraction of information regarding the possible behaviors of a system, with applicability to fluid mechanics and beyond, such as boundaries and transitions between qualitatively different kinds of behavior in data sets defined by meteorological, financial, psychological, or population observations. Education, outreach and dissemination are integral to the project activities, aimed at four audiences: (1) undergraduates will participate in research and develop learning materials for (2) high school students, who will learn about the role of chaotic transport and its relevance in the environment and biology; (3) graduate students and faculty will explore applications of tools from dynamical systems to new problem domains through a project-based course; and the (4) global community of researchers will benefit from numerical tools disseminated online which may reveal previously hidden structure in dynamic data sets.

该教师早期职业发展（职业）项目的研究目标是为相位空间运输问题的理论和计算分析奠定坚定的统一基础。可以将许多物理系统建模为具有潜在的动力骨架，该骨骼组织了所有可能的行为如何相关。对于以简单的时间依赖性进行分析的系统，该骨架由众所周知的数学结构（例如不变集和集合）组成，这些结构渐近地接近或偏离了此类集合。但是，对于基于数据的越来越多的兴趣的系统，它是非周期性的，并且仅在短时间内观察到，通常没有这样的结构。拟议的研究将找到对这些结构的适当类似物，以将动态系统结果的适用性扩展到现实世界数据。在流体流动的背景下，我们将使用几何，概率和拓扑方法开发工具，包括连贯的集合，几乎不变的集合和象征性动态。该项目的完善完成将提供一种新的和富有成果的方法，以实现概念化，可视化的信息，以及与系统相比，依赖于系统的范围，以及依赖于系统的范围，以及与系统的范围，以及综合性的范围，以及综合性的机制，以及流动性的流动性，流动性的机制，以及流动性的流动性，以及流动性的流动性，以及流动性的综合性。由气象，财务，心理或人口观察定义的数据集中的行为类型。教育，宣传和传播是针对四个受众的项目活动不可或缺的：（1）本科生将参加（2）高中生的研究和开发学习材料，他们将了解混乱的运输及其在环境和生物学中的相关性；（3）研究生和教职员工将通过基于项目的课程探索从动态系统到新问题领域的工具的应用；（4）全球研究人员社区将受益于在线传播的数值工具，这可能会揭示动态数据集中先前隐藏的结构。