Problems in Chaotic Dynamics

项目概述本计划是一个为期三年的混沌系统理论/计算研究计划。首席调查员和研究生将开展这项工作。预计，通过参与拟议的研究，研究生将精通非线性动力系统理论，计算机实验的设计和实施，以及物理系统的建模。将讨论两个问题：1)由许多相互关联的动态单元组成的系统的同步的开始：由许多相互关联的动态(可能是混沌)单元组成的系统的同步取决于单个连接单元的特性、耦合网络的拓扑结构以及沿每条链路的耦合的强度。我们过去在这个领域的工作是关于混沌单元系统，其中耦合是全局的(All-to-all)，并且所有的环节具有相同的耦合强度，并致力于提供从非相干行为到周期振荡的转换的一般理论。我们建议将这项工作大大扩展到更一般的连接拓扑。这个问题在物理和化学系统中很重要，但它最感兴趣的可能是生物系统，在生物系统中，相干振荡非常普遍，显然是许多小单位相互作用的结果。2)流体中的混沌混合和平流：这类一般问题对于大量的应用是极其重要的，但仍然存在非常有趣和重要的基本开放问题。我们以前的工作将有限时间Lyapunov指数的概念和大偏差理论引入到这一领域，并利用这一概念研究了各种流动情况下的分维、功率谱和结构函数。我们建议的主要研究领域将是受限混沌流动中刚性边界的影响(在大多数实验室实验中可能具有重要意义)。大的高度连接的系统的同步性的开始是一个内在的有趣的基本问题，并解决重要的现实世界问题，如生物系统中的振荡行为。混沌混合和平流是一个影响从大气科学到化学工程等领域的基本问题。对于这两个问题领域，国际和平研究所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在重要研究领域对研究生的培训，并通过与大学更大的混沌小组的互动，在这些问题上教育其他人。通过这项研究获得的理解将在包括物理、生物、工程和气象学在内的各种领域中有用。

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.