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

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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.