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.

该项目的主要目标是在分析和数值上研究一般动态系统的图。这项工作的结果将代表理解动态系统基本方面的重要进步。它将提供一个统一的设置和一组适用于任何动态系统的工具，从一维离散时间系统（例如逻辑图）到无限二维连续时间，例如Belousov-Zhabotinsky化学反应。该项目还旨在创建霍华德大学的一群研究生和本科生，从事对系统定性动态的数值分析以及主要研究人员。此外，在该项目中，调查人员将在参与学生的帮助下在物流地图上写一篇专着。该专着最终将在一个地方收集物流图上最重要的结果，并将针对应用读者，强调可读性而不是正式优雅。为了使最广泛的受众成为可能，该专着将以“开源”在线格式免费发布。该项目将在历史悠久的黑人研究大学霍华德大学全面开发和开发。该项目的主要目标是：1。研究几个有限维数和无限二维动力学系统的图，包括但不限于以下几个：单峰地图，多模式地图，Lorenz地图，强制倾倒悬浮板，Newhouse Maps，Newhouse Maps，newhouse Maps，semitear Parabolic Pdes.2。研究图形本身的一般特性，包括参数家族中节点的可能外观/消失的类型，图形连接的条件，节点和边缘的替代定义。此外，该项目将包括对几种复发概念的全面研究，例如链式发生，强大的链旋转和Auslander的广泛复发，以及开发一个公理系统，这些系统将作为各种复发的框架，并且可以从中可以证明动态系统的一般特性。数值结果将通过使用开发和用于数字研究逻辑图和Lorenz系统的高级版本来实现。将开发新的代码来研究与上观点中提到的半线性抛物线PDE相对应的无限尺寸系统，来自几种重要的化学反应 - 扩散现象的重要模型。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛的影响审查审查标准来通过评估来通过评估来获得支持的。