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

小行星9403691 本文研究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理论化学中的问题。 在过去五年中，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真实的时间动力学数据的程度，这导致了被称为"飞秒化学“的研究领域。 因此，我们正处于动力系统研究可以在解释这些新实验数据方面发挥作用的时刻。 许多感兴趣的问题是全球性的和几何的性质。 例如，对与分子内和分子间能量转移相关的问题的回答取决于与相空间中各种尺寸和形状的表面相关的几何形状和动力学。 有必要在这一领域的应用数学研究，因为大多数的理论化学家在这一领域的工作已经进行了低维模型。 由于大多数真实的分子模型都是高维的，因此发展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数学方法的发展，了解分子内和分子间的能量转移在更现实的分子系统。 本文研究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理论化学中的问题。 在过去5年中，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真实的时间动力学数据的程度，这导致了被称为"飞秒化学“的研究领域。 因此，我们正处于一个点，在这里，动力系统的研究可以在解释这些新的实验数据中发挥作用。 许多感兴趣的问题是全球性的和几何的性质。 例如，分子内和分子间能量转移相关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，这些不变流形在相空间中形成了控制能量转移问题的“网络”。 在这些区域附近，不变流形几何比标准的“不变托里”图复杂得多，需要开发新的方法。此外，人们经常遇到奇异摄动现象附近的这种共振区，迫使一个治疗“椭圆”和“双曲线”的现象同时进行。 一个很有前途的方法，这样的问题是所谓的“能量相位”的方法开发的Haller和Wiggins，主要是在上下文中的两个自由度的系统，它使一个连接在一起的“椭圆”和“双曲线”的现象，出现在共振附近。我们将把这种方法推广到多自由度系统。 与此同时，我们将有兴趣了解的机制，产生复杂的“混沌”的行为是“本质上高维”，即行为，这不仅仅是一个“放大”的版本典型的低维行为。 这项研究的结果之一将是数学方法的发展，了解分子内和分子间的能量转移在更现实的分子系统。

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.