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.

提案：DMS-0407110PI：Divakar Viswanathinstitution：密歇根大学Ann Arbortitle：低维动力学的符号动力学方法是低维混乱的数学描述中的一个重要和核心概念。符号动力学的主要原理是编码轨迹段的符号序列的结构。由于附近的混乱系统轨迹迅速差异，即使是较长的轨迹也将具有简短的符号代码。此外，这些符号代码明确表示轨迹家族之间的各种递归关系。 在较早的工作中，研究者使用符号动力学获得了众所周知的洛伦兹吸引子的分形结构的明确图，该图是1963年洛伦兹（Lorenz）推断出的，但在研究者工作之前没有明确表现出来。 研究者开发了数学和计算算法，将符号动力学方法应用于低维汉密尔顿系统。该项目的一个特殊重点是三体问题的物理有趣的实例。特别是，该项目研究了三体问题的共振转变，并观察了天体和航天器的运动。研究人员与其他研究人员合作，探讨了适用于具有低维吸引子的耗散部分微分方程的新方法。研究者使用该项目和相关研究的结果将研究的兴奋传递给本科和研究生的跨学科受众。批评性的变化，天气预测，行星和其他天体的运动，航天器轨迹的运动，航天器轨迹的运动，以及通过燃料电池的通道流过氢气，但要考虑一些不同的科学范围，这是一个不同的科学挑战。然而，所有这些不同的现象都是用称为非线性微分方程的相同类型的数学对象来描述的。数学的使用导致了可以阐明非常多样化现象的相当一般性的概念。一个示例是共振的概念，这意味着系统的一部分运动周期与另一部分的运动周期是一个简单的部分，例如1/2、2/1或1/1。 当系统的两个部分引起共鸣时，这两个部分的相互作用比其他导致意外结果的相互作用要强得多，其中一个意外的结果是丧失可预测性。 研究者研究与行星，卫星和航天器的运动有关的共鸣。研究人员开发了尽管可预测性丧失，但仍可以使准确的计算成为可能。 这些方法的关键是一个概念，它利用丧失可预测性来提供紧凑的变化和不可预测现象的描述。这个概念称为符号动态。