Graphs of Dynamical Systems

这个项目的主要目标是研究一般动力系统的图，包括解析和数值。这项工作的结果将代表对动力系统基本方面的理解的重要进展。它将提供一个统一的设置和一套工具，特别是适用于任何动力系统，从一维离散时间系统，如逻辑映射，到无限维连续时间系统，如别洛乌索夫-扎博廷斯基化学反应。该项目还旨在建立一个由霍华德大学的研究生和本科生组成的小组，与首席研究员一起对系统的定性动力学进行数值分析。此外，在这个项目中，研究者将在参与学生的帮助下撰写一篇关于物流地图的专著。这本专著最终将在一个地方收集逻辑地图上最重要的结果，并将针对应用读者，强调可读性而不是形式优雅。为了使它能够提供给尽可能广泛的受众，该专著将以“开源”在线格式免费发布。该项目将在霍华德大学（一所历史悠久的黑人研究型大学）全面开发和实施。这个项目的主要目标是：1。研究了几种有限维和无限维动力系统的图，包括但不限于：单峰图、多峰图、Lorenz图、强迫倾倒摆、Newhouse图、半线性抛物型pdes。研究图本身的一般性质，包括参数族中节点可能出现/消失的类型，图连接的条件，节点和边的替代定义。此外，本项目将包括对几个递归概念的综合研究，如链式递归、强链式递归和Auslander的广义递归，并建立一个公理体系，作为各种递归的框架，并以此为基础证明动力系统图的一般性质。数值结果将通过使用开发和用于数值研究逻辑图和洛伦兹系统的代码的改进版本来实现。将开发新的代码来研究与前一点提到的半线性抛物偏微分方程相对应的无限维系统，这些系统来自几个重要的化学反应扩散现象模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main goal of this project is to study graphs of general dynamical systems, both analytically and numerically. The outcome of this work will represent an important advance in the understanding of fundamental aspects of dynamical systems. It will provide a unifying setting and a set of tools applicable, in particular, to any dynamical system, from one-dimensional discrete-time systems, such as the logistic maps, to infinite-dimensional continuous-time ones, such as the Belousov–Zhabotinsky chemical reaction. This project also aims at the creation of a group of graduate and undergraduate students at Howard University working on the numerical analysis of the qualitative dynamics of the systems, together with the lead investigator. Moreover, within this project, the investigator will write a monograph on the logistic map, with help from participating students. This monograph will finally collect the most important results on the logistic map in a single place and will be aimed at applied readers, emphasizing readability over formal elegance. In order to make it available to the widest audience possible, the monograph will be released freely in “open source” online format. The project will be fully developed and undertaken at Howard University, a Historically Black Research University. The main goals of this project are: 1. Investigating the graph of several finite-dimensional and infinite-dimensional dynamical systems, including but not limited to the following: unimodal maps, multimodal maps, Lorenz map, forced dumped pendulum, Newhouse maps, semilinear parabolic PDEs.2. Investigating the general properties of graphs themselves, including the types of possible appearance/disappearance of nodes in parametric families, the conditions for the graph to be connected, alternate definitions of nodes and edges. Moreover, this project will include a comprehensive study of several concepts of recurrence, such as chain-recurrence, strong chain-recurrence and Auslander’s generalized recurrence, and developing a system of axioms that will work as a framework for all kinds of recurrence and from which it will be possible to prove general properties of graphs of dynamical systems. The numerical results will be achieved by using refined versions of the codes developed and used to numerically study the logistic map and the Lorenz system. New code will be developed to study the infinite dimensional systems corresponding to the semilinear parabolic PDEs mentioned in the previous point, coming from several important models of chemical reaction-diffusion phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.