Symbolic Dynamics Methods for Low Dimensional Dynamics

低维动力学的符号动力学方法

• 批准号: 0407110
• 负责人: Divakar Viswanath
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0407110&HistoricalAwards=false
• 关键词: Symbolic; Dynamics; Methods; Low; Dimensional

Proposal: DMS-0407110PI: Divakar ViswanathInstitution: University of Michigan Ann ArborTitle: Symbolic dynamics methods for low-dimensional dynamicsABSTRACTSymbolic dynamics is an important and central concept in the mathematical description of low-dimensional chaos. The leading principle of symbolic dynamics is the construction of symbol sequences that encode segments of trajectories. As nearby trajectories of a chaotic system diverge rapidly, even long segments of trajectories will have short symbolic codes. Furthermore, these symbolic codes explicitly indicate various recursive relationships between families of trajectories. In earlier work, the investigator used symbolic dynamics to obtain explicit plots of the fractal structure of the well-known Lorenz attractor, which was inferred by Lorenz in 1963 but was not exhibited explicitly prior to the investigator's work. The investigator develops mathematics and computational algorithms to apply the symbolic dynamics method to low dimensional Hamiltonian systems. A particular focus of this project is on physically interesting instances of the three-body problem. In particular, this project studies resonance transitions in the three-body problem with a view towards the motion of celestial bodies and spacecrafts. In collaboration with other researchers, the investigator explores new methods that apply to dissipative partial differential equations with low dimensional attractors. The investigator uses results from this project and related research to transmit the excitement of research to an interdisciplinary audience of undergraduate and graduate students.Climactic changes, weather prediction, the motion of planets and other celestial bodies, spacecraft trajectories, and the flow of hydrogen through the channels of a fuel cell, to consider but a few examples, present quite different scientific challenges. Yet all these diverse phenomena are described by the same type of mathematical objects called nonlinear differential equations. The use of mathematics leads to concepts of quite great generality that can illuminate very diverse phenomena. An example is the concept of resonance, which means that the ratio of the period of motion of one part of the system to that of another part is a simple fraction such as 1/2, 2/1, or 1/1. When two parts of a system are in resonance, the two parts interact much more strongly than otherwise leading to unexpected results, one of these unexpected results being the loss of predictability. The investigator studies resonances related to the motion of planets, satellites, and spacecrafts. The investigator develops methods that make accurate computations possible in spite of the loss of predictability. The key to these methods is a concept that utilizes the loss of predictability to give compact descriptions of changing and unpredictable phenomena. This concept is called symbolic dynamics.

提议：DMS-0407110PI：Divakar ViswanathInstitution：密歇根大学Ann Arborle标题：低维动力学的符号动力学方法摘要符号动力学是低维混沌数学描述中的一个重要而核心的概念。符号动力学的主要原理是构建对轨迹段进行编码的符号序列。随着一个混沌系统附近的轨迹迅速偏离，即使是长的轨迹段也会有短的符号代码。此外，这些符号代码明确表示轨迹系列之间的各种递归关系。在早期的工作中，研究人员使用符号动力学来获得著名的Lorenz吸引子的显式图形，这是由Lorenz在1963年推断的，但在研究人员的工作之前并没有显式展示。研究人员发展了数学和计算算法，将符号动力学方法应用于低维哈密顿系统。这个项目的一个特别的焦点是三体问题的物理有趣的例子。特别是，该项目研究了三体问题中的共振跃迁，以期实现天体和航天器的运动。与其他研究人员合作，研究人员探索了适用于具有低维吸引子的耗散偏微分方程解的新方法。研究人员利用这个项目和相关研究的结果，将研究的兴奋传递给本科生和研究生的跨学科受众。气候变化、天气预报、行星和其他天体的运动、航天器轨迹以及氢通过燃料电池通道的流动，仅举几个例子，提出了相当不同的科学挑战。然而，所有这些不同的现象都是由同一类型的数学对象描述的，称为非线性微分方程式。数学的使用导致了具有相当大普遍性的概念，这些概念可以解释非常不同的现象。一个例子是共振的概念，它意味着系统的一个部分与另一个部分的运动周期之比是一个简单的分数，如1/2、2/1或1/1。当系统的两个部分处于共振状态时，这两个部分的相互作用比其他情况下更强烈，从而导致意想不到的结果，其中一个意想不到的结果是失去可预测性。研究人员研究与行星、卫星和宇宙飞船运动有关的共振。研究人员开发了一些方法，使准确的计算成为可能，尽管失去了可预测性。这些方法的关键是一个概念，它利用可预测性的损失对变化和不可预测的现象进行紧凑的描述。这个概念被称为符号动力学。