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

在应用科学的许多领域，人们被引导去研究依赖于一个或多个参数的动力系统。除了初始瞬态外，状态向量通常接近于决定动力学行为的低维不变流形。研究者开发并分析了路径跟随过程中这种不变流形的直接数值计算方法。此外，我们还研究了如何检测和分类潜在的分支。到目前为止，只有当不变流形为不动点或周期轨道时，才存在较好的路径跟踪和分岔检测算法。下一个有趣的例子是不变2环面的分支，典型地对应于具有两个频率的准周期运动。在之前的工作中，研究者和他的合作者已经开发了一种可靠的代码来计算2-环面的分支，但是对它们的分支的数值研究才刚刚开始。这个项目在几个方向上扩展了以前的工作，最重要的是以下几点。首先，在代码中添加一些指标，可以判断是否可能出现分岔，并且——理想情况下——可以判断分岔的性质。其次，用局部图代替全局图来表示不变流形。该扩展使局部网格细化成为可能，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，科学家们已经发展出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、纳维耶和斯托克斯开发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式是最近才发展起来的。虽然这些方程在原则上决定了进化，但通常需要超级计算机来实际评估这些预测。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通常也不足以进行完整的建模。那么就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）随着时间的推移稳定在状态空间中的低维对象，则这些是可能的。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，从应用力学到化学再到材料科学。另一个目的是对状态向量逼近的低维几何对象进行直接数值计算。这个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.