Mathematical Sciences: Global Dynamics and Geometry in High Dimensional Nonlinear Dynamical Systems

9403691维格斯这项研究涉及高维非线性动力系统的全局几何分析。许多分析的物理动机来自于理论化学中的问题。在过去的五年里，化学实验技术已经发展到可以获得与分子相互作用和动力学有关的实时动态数据的程度，这导致了被称为“飞秒化学”的研究领域。因此，我们正处在一个点上，动力系统研究可以在解释这些新的实验数据方面发挥作用。许多令人感兴趣的问题本质上是全球性和几何性的。例如，与分子内和分子间能量转移有关的问题的答案取决于与相空间中不同维度和形状的表面相关联的几何和动力学。由于理论化学家在这一领域的大部分工作都是用低维模型进行的，因此需要在这一领域进行应用数学研究。由于大多数现实的分子模型都是高维的，因此开发适用于高维系统的数学技术以及理解一般的高维动力学现象是很重要的。这项研究的结果之一将是发展数学方法来理解更现实的分子系统中的分子内和分子间的能量转移。这项研究涉及高维非线性动力系统的全局几何分析。许多分析的物理动机来自于理论化学中的问题。在过去的5年里，化学实验技术已经发展到可以获得与分子相互作用和动力学有关的实时动态数据的程度，这导致了被称为“飞秒化学”的研究领域。因此，我们在这里的一个点上，动力系统研究可以在解释这些新的实验数据中发挥作用。许多令人感兴趣的问题本质上是全球性和几何性的。例如，与分子内和分子间能量转移有关的问题的答案取决于与相空间中共振附近出现的不变流形相关的几何和动力学，而这些不变流形在相空间中形成了支配能量转移问题的“网络”。在这样的区域附近，不变流形几何比标准的不变圆环图复杂得多，需要开发新的方法。此外，在这种共振区附近经常会遇到奇异摄动现象，这迫使人们同时处理“椭圆”和“双曲”现象。解决这类问题的一个有希望的方法是由Haller和Wiggins开发的所谓的“能量相”方法，主要是在两自由度系统的背景下，它使人们能够将在共振附近出现的“椭圆”和“双曲线”现象结合在一起。我们将把这种方法推广到多自由度系统。同时，我们将感兴趣的是理解导致复杂的“混乱”行为的机制，这些行为是“本质上高维的”，即不只是典型的低维行为的“放大”版本。这项研究的结果之一将是发展数学方法来理解更现实的分子系统中的分子内和分子间的能量转移。

9403691 Wiggens This research is concerned with the global, geometric analysis of high dimensional nonlinear dynamical systems. The physical motivation for much of the analysis arises from problems in theoretical chemistry. Over the past five years experimental techniques have been developed in chemistry to the point where real time dynamical data related to molecular interactions and dynamics can be obtained, which has resulted in the area of research known as ``femtochemistry''. As a result, we are at a point where dynamical systems research can play a role in the interpretation of this new experimental data. Many of the questions of interest are global and geometrical in nature. For example, answers to questions related to intramolecular and intermolecular energy transfer depend on the geometry and dynamics associated with surfaces of various dimensions and shapes in phase space. There is a need for applied mathematical research in this area since most of the work of theoretical chemists in this area has been carried out with low dimensional models. Since most realistic models of molecules are higher dimensional, it is important to develop mathematical techniques that apply to high dimensional systems as well as understand higher dimensional dynamical phenomena in general. One result of this research will be the development of mathematical methods for understanding intramolecular and intermolecular energy transfer in more realistic molecular systems. This research is concerned with the global, geometric analysis of high dimensional nonlinear dynamical systems. The physical motivation for much of the analysis arises from problems in theoretical chemistry. Over the past 5 years experimental techniques have been developed in chemistry to the point where real time dynamical data related to molecular interactions and dynamics can be obtained, which has resulted in the area of research known as ``femtochemistry''. As a result, we are at a point w here dynamical systems research can play a role in the interpretation of this new experimental data. Many of the questions of interest are global and geometrical in nature. For example, answers to questions related to intramolecular and intermolecular energy transfer depend on the geometry and dynamics associated with invariant manifolds that arise near resonances in phase space, and these invariant manifolds form the ``network'' in phase space which governs energy transfer issues. Near such regions the invariant manifold geometry is much more complicated than the standard ``invariant tori'' picture and new methods need to be developed. Also, one often encounters singular perturbation phenomena near such resonance regions which forces one to treat ``elliptic'' and ``hyperbolic'' phenomena simultaneously. One promising method for such problems is the so-called ``energy-phase'' method developed by Haller and Wiggins, largely in the context of two-degree-of-freedom systems, which enables one to join together the ``elliptic'' and ``hyperbolic'' phenomena that arises near resonances. We will extend this method to multi-degree-of-freedom systems. At the same time we will be interested in understanding mechanisms that give rise to complicated ``chaotic'' behavior that are ``intrinsically high dimensional'', i.e. behavior that is not just a ``scaled up'' version of typical low dimensional behavior. One result of this research will be the development of mathematical methods for understanding intramolecular and intermolecular energy transfer in more realistic molecular systems.