Mathematical Sciences: "Geometric Methods and Their Applications for Multi-Dimensional Resonant Dynamical Systems

9501239哈勒共振是动力系统相空间中某些角变量的频率几乎相等的区域。这一建议旨在发展和应用新的几何方法来研究复杂的近共振动力学。主要目标是捕获相空间中对系统总体演化有强烈影响的不变结构。这种结构的理想候选者是在许多共振问题中出现的双曲部分慢流形，并且在不同的时间尺度上进行物理上重要的运动。正则和奇异摄动技术的微妙结合可以通过检测连接这些流形邻域的显著族解来详细描述这种效应。这样的解描述了不同物理运动之间重复的、复杂的过渡。自然界中的许多进化过程都是微分方程组的解。这些解的集合可以可视化为一些高维相空间中的曲面，而进化系统不同状态之间的不规则转变则表现为连接这些曲面的混沌轨道。当解表面由几个不同时间尺度的解组成时，该方法处理了在重要且频繁的情况下对这种过渡的检测。提出的数学技术可用于研究尚未解释的分子振动细节，海洋动力学中表面波的模式，行星系统的引力相互作用以及工程结构的混沌振荡。

9501239 Haller Resonances are regions in the phase space of a dynamical system in which the frequencies of some angular variables become nearly commensurate. This proposal is aimed toward the development and application of new geometric methods for the study of complicated dynamics near resonances. The main goal is to capture invariant structures in the phase space with a strong impact on the general evolution of the system. Ideal candidates for such structures are hyperbolic partially slow manifolds that arise in many resonance problems and carry physically important motions with different time scales. A subtle combination of regular and singular perturbation techniques can describe this effect in detail by detecting remarkable families of solutions connecting neighborhoods of these manifolds. Such solutions describe repeated, complicated transitions between physically distinguished motions Many evolutionary processes in nature as solutions of sets of differential equations. Sets of these solutions can be visualized as surfaces in some higher dimensional phase space and irregular transitions between different states of the evolutionary system appear as chaotic orbits connecting these surfaces. This proposal deals with the detection of such transitions in the important and frequent case when the solution surfaces are composed of solutions with several different time scales. The proposed mathematical techniques can be used to study yet unexplained details of molecular vibrations, patterns of surface waves in ocean dynamics, gravitational interactions of planetary systems, and chaotic oscillations of engineering structures.