Graphs of Dynamical Systems

动力系统图

• 批准号: 2308225
• 负责人: Roberto De Leo
• 金额: $ 29.29万
• 依托单位: Howard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2308225&HistoricalAwards=false
• 关键词: Graphs; Dynamical; Systems

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.

这个项目的主要目标是研究一般动力系统的图形，包括解析和数值两方面。这项工作的结果将代表着在理解动力系统基本方面的重要进展。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Logistic映射，到无限维连续时间系统，如Belousov-Zhabotinsky化学反应。该项目还旨在创建一组霍华德大学的研究生和本科生，与首席研究员一起从事系统定性动力学的数值分析工作。此外，在这个项目中，调查员将在参与的学生的帮助下，在逻辑地图上写一本专著。这本专著将最终在一个地方收集物流地图上最重要的结果，并将针对应用读者，强调可读性而不是正式的优雅。为了让尽可能多的读者能够阅读到这本专著，这本专著将以“开源”的在线格式免费发布。该项目将在霍华德大学全面开发和实施，霍华德大学是一所历史上的黑人研究型大学。本项目的主要目标是：1.研究几个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、洛伦兹映射、强迫抛物摆、纽豪斯映射、半线性抛物型PDES。研究图本身的一般性质，包括参量族中节点可能出现/消失的类型、图的连通条件、节点和边的替代定义。此外，这个项目将包括对几个递归概念的全面研究，如链-递归、强链-递归和Auslander广义递归，并开发一个公理系统，它将作为所有类型递归的框架，并将可能从它证明动力系统图的一般性质。数值结果将通过使用开发的改进版本的代码来实现，并用于对Logistic映射和Lorenz系统进行数值研究。将开发新的代码来研究与前面提到的半线性抛物型偏微分方程组相对应的无限维系统，该代码来自几个重要的化学反应-扩散现象模型。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。