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这项研究与高维非线性动力学系统的全局几何分析有关。 大部分分析的物理动机来自理论化学中的问题。 在过去的五年中，实验技术是在化学中开发的，即可以获得与分子相互作用和动力学相关的实时动力学数据，这导致了被称为``fem fem tem tem fem tem fem tem fem'''的研究领域。 结果，我们正处于动态系统研究中可以在解释这一新实验数据中发挥作用。 许多感兴趣的问题本质上都是全球和几何问题。 例如，与分子内和分子间的能量转移有关的问题的答案取决于与相位空间中各个维度和形状的表面相关的几何和动力学。 由于该领域的大多数理论化学家的工作都是使用低维模型进行的，因此需要在该领域进行数学研究。 由于大多数现实的分子模型是较高的维度，因此开发适用于高维系统的数学技术以及一般而言的高维动力学现象非常重要。这项研究的结果将是发展数学方法，以理解更现实的分子系统中分子内和分子间的能量转移。 这项研究与高维非线性动力学系统的全局几何分析有关。 大部分分析的物理动机来自理论化学中的问题。 在过去的五年中，实验技术是在化学中开发的，即可以获得与分子相互作用和动力学有关的实时动态数据，这导致了被称为``fem femto'''的研究领域。 结果，我们处于动态系统研究可以在解释这一新实验数据中发挥作用。 许多感兴趣的问题本质上都是全球和几何问题。 例如，对与分子内和分子间的能量转移有关的问题的答案取决于与不变的流形相关的几何和动态，这些歧管在相位空间中产生几乎谐振，并且这些不变的歧管在控制能量传递问题的相空间中形成了“网络” 。 在这样的区域附近，不变的歧管几何形状比标准的``不变式''图片要复杂得多，需要开发新的方法。同样，人们经常在这样的共振区域附近遇到奇异的扰动现象，这迫使人们同时对待``椭圆形''和``双曲线''现象。 解决此类问题的一种有前途的方法是Haller和Wiggins开发的所谓``能量相''方法，主要是在两度自由的系统的背景下，这使人们能够将``椭圆形''一起连接在一起。 '和``双曲线''现象近在共鸣。我们将将此方法扩展到多度自由度系统。 与此同时行为。 这项研究的结果将是发展数学方法，以理解更现实的分子系统中分子内和分子间的能量转移。