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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.

本学院早期职业发展（CAREER）项目的研究目标是为相空间传输问题的理论和计算分析奠定坚实的统一基础。许多物理系统可以被建模为具有一个底层的动力学骨架，该骨架组织所有可能的行为如何相关。对于具有简单时间依赖性的解析已知系统，这个骨架由众所周知的数学结构组成，例如不变集和渐近接近或偏离这些集的集。但是对于越来越感兴趣的非周期性系统，基于数据，并且仅在短时间内观察到，通常不存在这样的结构。拟议的研究将找到适当的类似这些结构，以扩大适用性的动力系统的结果，真实的世界的数据。在流体流动的背景下，我们将开发使用几何，概率和拓扑方法的工具，包括相干集，几乎不变集和符号动力学。该项目的成功完成将为概念化，可视化和提取有关系统可能行为的信息提供一种新的和富有成效的方法，适用于流体力学及其他领域，例如在由气象、金融、心理或人口观测定义的数据集中的定性不同种类的行为之间的边界和过渡。教育、外联和传播是项目活动的组成部分，针对四个受众：（1）本科生将参与研究并为他们编写学习材料;（2）高中生将了解混沌运输的作用及其在环境和生物学中的相关性;（3）研究生和教师将通过项目课程探索从动力系统到新问题领域的工具应用;以及（4）全球研究人员将受益于在线传播的数字工具，这些工具可能揭示动态数据集中先前隐藏的结构。