Mathematical Sciences: Computation and Analysis of Invariant Manifolds and Their Bifurcations

在应用科学的许多领域中，人们会研究依赖于一个或多个参数的动力系统。 状态向量通常接近一个低维不变流形，它决定了动力学行为，除了初始瞬态。 研究人员开发和分析的方法，直接数值计算这样的不变流形的路径跟踪过程。 此外，我们研究了如何检测和分类潜在的分叉。 到目前为止，发展良好的算法路径跟踪和检测的分歧存在时，不变流形是一个固定点或周期轨道。 下一个有趣的情况是不变2环面的分支，通常对应于具有两个频率的准周期运动。 在以前的工作中，调查员和他的合作者已经开发了一个可靠的代码来计算这样的分支2环面，但其分叉的数值研究才刚刚开始。 该项目在几个方向上扩展了以前的工作，最重要的是以下方面。 首先，在代码中添加指示符，可以判断是否可能出现分叉，并且--理想情况下--可以判断分叉的性质。 其次，为了表示不变流形，使用局部图而不是一个全局图。 这种扩展使得局部网格细化变得实用，并大大提高了现有代码的性能和适用性。 在过去的四个世纪里，科学家们已经开发出了各种方程来确定各种系统的时间演化。 牛顿写下了行星运动的方程式。 欧拉、纳维尔和斯托克斯发展了方程来预测水和其他流体的流动。 控制气象学、海洋学或物质相变的方程是最近才发展起来的。 虽然这些方程在原则上决定了进化，但超级计算机通常需要实际评估预测。 当然，原因是所涉及的过程非常复杂。 事实上，即使是最快的计算机也往往不足以进行完整的建模。 因此，有必要要求合理的简化。 如果状态向量（描述系统在给定时刻的状态）随着时间的推移而稳定到状态空间中的低维对象，则这些都是可能的。 这项研究的目的之一是了解这种行为何时发生。 有许多已知的例子，从应用力学到化学到材料科学。 另一个目的是直接数值计算的低维几何对象，接近的状态向量。 该项目有助于更好地理解动力系统。 反过来，这导致了更好的算法来预测进化，并使更广泛的过程和现象适合超级计算机建模。

In many areas of the applied sciences one is led to study dynamical systems depending on one or more parameters. The state vector often approaches a low-dimensional invariant manifold that determines the dynamical behavior, except for an initial transient. The investigator develops and analyszes methods for the direct numerical computation of such invariant manifolds in a path-following process. Also, we studies how to detect and classify potential bifurcations. So far, well-developed algorithms for path following and for detection of bifurcations exist only when the invariant manifold is a fixed point or a periodic orbit. The next interesting cases are branches of invariant 2-tori, corresponding typically to quasiperiodic motion with two frequencies. In previous work the investigator and his collaborators have developed a reliable code to compute such branches of 2-tori, but the numerical study of their bifurcations has just begun. This project extends the previous work in several directions, the most important being the following. First, indicators are added to the code that can tell if a bifurcation is possible, and -- ideally -- can tell the nature of the bifurcation. Second, to represent the invariant manifold, local charts are used instead of one global chart. This extension makes local mesh refinement practical and greatly enhances the performance and applicability of the existent code. During the last four centuries scientists have developed equations to determine the time evolution of a great variety of systems. Newton wrote down the equations for planetary motion. Euler, Navier, and Stokes developed equations to predict the flow of water and other fluids. The equations governing meteorology, oceanography, or the phase changes of materials were developed more recently. Though the equations determine the evolution in principle, supercomputers are usually necessary to actually evaluate the predictions. The reason is, of course, the great complexity of th e processes involved. In fact, even the fastest computers are often insufficient for a full modeling. Then it is necessary to ask for sensible simplifications. These are possible if the state vector (which describes the state of the system at a given instant of time) settles to a low-dimensional object in state space as time progresses. It is one aim of the proposed research to understand when such a behavior occurs. There are many known examples, ranging from applied mechanics to chemistry to material sciences. Another aim is the direct numerical computation of the low-dimensional geometric object that is approached by the state vector. The project helps understand dynamical systems better. In turn, this leads to better algorithms for predicting evolutions and makes a wider range of processes and phenomena amenable to modeling by supercomputers.