Graphs of Dynamical Systems

这个项目的主要目标是研究一般动力系统的图形，包括分析和数值。这项工作的结果将代表一个重要的进展，在了解动力系统的基本方面。它将提供一个统一的设置和一套工具，特别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Belousov-Zhabotinsky化学反应。该项目还旨在在霍华德大学建立一个研究生和本科生小组，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，研究人员将在参与学生的帮助下撰写一本关于逻辑地图的专著。这本专着将最终收集在一个单一的地方，并将在逻辑地图上的最重要的结果，旨在应用读者，强调可读性超过正式优雅。为了使尽可能多的读者能够获得，该专著将以“开放源码”的在线格式免费发行。该项目将在霍华德大学全面开发和开展，这是一所历史上的黑人研究大学。本项目的主要目标是：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：单峰映射，多峰映射，Lorenz映射，强迫倾倒摆，纽豪斯映射，半线性抛物偏微分方程.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，图被连接的条件，节点和边的替代定义。 此外，该项目将包括对递归的几个概念进行全面研究，例如链递归、强链递归和Auslander的广义递归，并开发一个公理系统，该公理系统将作为各种递归的框架，并有可能证明动力系统图形的一般性质。数值结果将通过使用改进版本的代码开发和使用数值研究的逻辑映射和洛伦兹系统。新的代码将被开发来研究对应于半线性抛物偏微分方程的无限维系统在前一点中提到的，来自化学反应扩散现象的几个重要模型。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.