Dynamics at a fixed resolution

这个项目的重点是扩展和优化动力系统计算研究的现有技术。认识到需要对动力系统进行离散化和截断作为进行数值研究的第一步，研究人员和她的合作者将重点放在可以克服相应信息损失的技术上。特别是，在构造用于数值计算的系统的有限表示时，它们包含了离散化、截断和在此过程中积累的其他误差的显式界限。有限表示是以固定分辨率查看的动力系统--关于各个点的精确位置及其轨迹的信息丢失，但保留粗略的拓扑结构。研究人员将结合拓扑工具，如代数拓扑学和Conley指数理论，在有限表示的计算中检测和证明原始系统动力学的存在。动力系统模型在整个社会中被使用。一些例子包括用于飓风预测的天气模型和用于研究环境对人口数量和持久性的影响的人口模型。目前，许多研究人员使用高性能的计算机模拟和统计技术来研究这样的动力系统。在光谱的另一端，数学家已经能够在更抽象的数学模型中破译高度复杂的动力学。本提案中描述的工作旨在成为这两种方法之间的桥梁。更具体地说，这位研究人员和她的合作者专注于计算技术的开发，这些技术使用复杂的数学工具并产生数学上严格的结果。这些数学工具来自代数拓扑学、分析、数值分析和动力系统理论等领域，可以用来解释所研究系统中一些感兴趣的现象。以前在种群生态学和热对流模型中研究复杂动力学的进展推动了这些继续研究的进行。

This project focuses on extending and optimizing existing techniques for computational studies of dynamical systems. Recognizing the need for discretizations and truncations of dynamical systems as a first step towards making a numerical study, the investigator and her collaborators focus on techniques that can overcome the corresponding loss of information. In particular, in constructing the finite representation of the system used for numerical computations, they incorporate explicit bounds for discretization, truncation, and other errors accumulated during this process. The finite representation is the dynamical system viewed at a fixed resolution -- information about the precise location of individual points and their trajectories is lost, but coarse, topological structures remain. The investigator will incorporate topological tools, such as algebraic topology and Conley index theory, in computations on the finite representation to detect and prove the existence of dynamics for the original system.Dynamical systems models are being used throughout society. Some examples include weather models used for hurricane prediction and population models used to study environmental effects on population size and persistence. Currently, many researchers study dynamical systems like these using high powered computer simulations and statistical techniques. On the other end of the spectrum, mathematicians have been able to decipher highly complicated dynamics in more abstract mathematical models. The work described in this proposal aims to serve as a bridge between these two approaches. More specifically, the investigator and her collaborators focus on the development of computational techniques that use sophisticated mathematical tools and yield mathematically rigorous results. The mathematical tools come from the fields of algebraic topology, analysis, numerical analysis, and dynamical systems theory and may be used to decipher some of the phenomena of interest in the studied systems. Prior progress in studying complicated dynamics in models from population ecology and heat convection motivates these continued studies.