Problems in Chaotic Dynamics

混沌动力学问题

• 批准号: 0456240
• 负责人: Edward Ott
• 金额: $ 25.37万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456240&HistoricalAwards=false
• 关键词: Problems; Chaotic; Dynamics

Project Summary This proposal is for a three-year theoretical/computational program on the study of chaotic systems. The principal investigator and graduate students will carry out the work. It is anticipated that, through their participation in the proposed research, the graduate students will become proficient in the theory of nonlinear dynamical systems, the design and implementation of computer experiments, and the modeling of physical systems. Two problem areas will be addressed: 1) The onset of synchronization in systems consisting of many interconnected dynamical units: Synchronization in systems of many interconnected dynamical (perhaps chaotic) units depends on the characteristics of the individual connected units, on the topology of the coupling network, and on the strengths of the couplings along each link. Our past work in this area was on systems of chaotic units where the coupling was global (all-to-all) and all links had equal coupling strength, and focused on providing a general theory of the transition from incoherent behavior to periodic oscillation. We propose to greatly extend this work to much more general connection topologies. This problem is important in physical and chemical systems, but perhaps its greatest interest is for biological systems where coherent oscillations are extremely prevalent and apparently result from the interaction of many small units. 2) Chaotic mixing and advection in fluids: This general class of problems is extremely important for a large variety of applications, yet there remain very interesting and significant basic open problems. Our past work has introduced the concept of finite time Lyapunov exponents and large deviation theory to this area, and we have used this to study fractal dimension, power spectra, and structure functions in a variety of flow situations. The main area of our proposed investigation will be on the effect of rigid boundaries in confined chaotic flows (potentially significant in most laboratory experiments). Intellectual Merit The onset of synchronism in large highly connected systems is a basic problem of inherent interest, and addresses important real-world problems, such as oscillatory behavior in biological systems. Chaotic mixing and advection is a problem of fundamental importance with impact in fields ranging from atmospheric science to chemical engineering. For both problem areas, previous work by the PI will serve as a strong base and starting point for the proposed research. Broader Impacts The proposed activity will promote the training of graduate students in important areas of research, and, by interaction with the larger chaos group at the university, it will educate others in these issues. The understanding gained through this research will be useful in a variety of fields, including physics, biology , engineering, and meteorology.

本项目是一个为期三年的混沌系统理论/计算研究项目。主要研究者和研究生将开展这项工作。预计，通过他们在拟议的研究参与，研究生将成为精通非线性动力系统的理论，计算机实验的设计和实施，以及物理系统的建模。两个问题领域将被解决：1）在系统中的同步的开始，由许多相互关联的动态（也许是混沌）单元的系统中的同步取决于单个连接的单元的特性，在耦合网络的拓扑结构，以及在强度的耦合沿着每个链接。我们过去在这方面的工作是在系统的混沌单位的耦合是全球性的（所有对所有）和所有的链接有相等的耦合强度，并专注于提供一个一般理论的过渡，从非相干行为的周期性振荡。我们建议大大扩展这项工作更一般的连接拓扑结构。这个问题在物理和化学系统中很重要，但它最大的兴趣可能是生物系统，其中相干振荡非常普遍，显然是由许多小单元的相互作用引起的。2)流体中的混沌混合和平流：这类问题对于各种各样的应用都非常重要，但仍然存在非常有趣和重要的基本开放问题。我们过去的工作将有限时间李雅普诺夫指数和大偏差理论的概念引入到该领域，并利用它来研究各种流动情况下的分维、功率谱和结构函数。我们提出的调查的主要领域将是在有限的混沌流（在大多数实验室实验中可能显着）的刚性边界的效果。在大型高度连接的系统中同步的开始是一个基本的问题，它解决了重要的现实问题，例如生物系统中的振荡行为。混沌混合和平流是一个重要的基本问题，在从大气科学到化学工程的各个领域都有影响。对于这两个问题领域，PI以前的工作将作为拟议研究的坚实基础和起点。更广泛的影响拟议的活动将促进研究生在重要研究领域的培训，并通过与大学中更大的混乱集团的互动，它将在这些问题上教育其他人。通过这项研究获得的理解将在物理学，生物学，工程学和气象学等各个领域都很有用。