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

数学科学：高维非线性动力系统中的全局动力学和几何

• 批准号: 9403691
• 负责人: Stephen Wiggins
• 金额: $ 5万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403691&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Global; Dynamics; Geometry

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.

9403691 Wiggens 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源于理论化学问题。 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相关的实时动态数据的程度，这导致了被称为“飞化学”的研究领域。 因此，我们正处于动力系统研究可以在解释这一新实验数据方面发挥作用的时刻。 许多令人感兴趣的问题本质上都是全局性的和几何性的。 例如，与分子内和分子间能量转移相关的问题的答案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。 由于理论化学家在该领域的大部分工作都是使用低维模型进行的，因此需要在该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非常重要。这项研究的成果之一将是开发数学方法来理解更现实的分子系统中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源于理论化学问题。 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相关的实时动态数据的程度，这导致了被称为“飞化学”的研究领域。 因此，我们正处于动力系统研究可以在解释这一新实验数据方面发挥作用的时刻。 许多令人感兴趣的问题本质上都是全局性的和几何性的。 例如，与分子内和分子间能量转移相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形状和动力学，并且这些不变流形在相空间中形成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“不变环面”图复杂得多，需要开发新方法。此外，人们经常在这些共振区域附近遇到奇异的扰动现象，这迫使人们同时处理“椭圆”和“双曲”现象。 解决此类问题的一种有前途的方法是由 Haller 和 Wiggins 开发的所谓“能量相”方法，主要是在二自由度系统的背景下，该方法使人们能够将“椭圆”连接在一起”和在共振附近出现的“双曲”现象。我们将把这种方法扩展到多自由度系统。 同时，我们将有兴趣理解产生“本质上高维”的复杂“混乱”行为的机制，即行为不仅仅是典型低维的“放大”版本行为。 这项研究的成果之一将是开发数学方法来理解更现实的分子系统中分子内和分子间的能量转移。