Symbolic Dynamics Methods for Low Dimensional Dynamics

建议：DMS-0407110 PI：Divakar Viswanath机构：密歇根大学安娜堡标题：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和核心概念。符号动力学的主要原理是构造对轨迹段进行编码的符号序列。由于混沌系统附近的轨迹迅速发散，即使是很长的轨迹段也会有很短的符号代码。此外，这些符号代码明确地表示轨迹族之间的各种递归关系。 在早期的工作中，研究者使用符号动力学来获得著名的洛伦兹吸引子的分形结构的显式图，这是洛伦兹在1963年推断的，但在研究者的工作之前没有明确展示。 研究人员开发数学和计算算法，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重点是在物理上有趣的例子的三体问题。特别是，该项目研究三体问题中的共振跃迁，以期研究天体和航天器的运动。与其他研究人员合作，研究人员探索适用于低维吸引子耗散偏微分方程的新方法。研究人员利用该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观众。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，提出了完全不同的科学挑战。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。数学的使用导致了具有相当普遍性的概念，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，这意味着系统的一部分的运动周期与另一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。 当一个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要强烈得多，从而导致意想不到的结果，这些意想不到的结果之一就是可预测性的丧失。 研究员研究与行星、卫星和航天器运动有关的共振。研究人员开发的方法，使准确的计算可能，尽管损失的可预测性。 这些方法的关键是一个概念，利用可预测性的损失，给变化和不可预测的现象紧凑的描述。这个概念被称为符号动力学。

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.