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

9501239哈勒共振是动力学系统相空间中某些角变量的频率变得近乎公度的区域。这一建议旨在发展和应用新的几何方法来研究共振附近的复杂动力学。主要目标是捕捉相空间中的不变结构，这些结构对系统的一般演化具有强烈的影响。这种结构的理想候选者是双曲部分慢流形，它们出现在许多共振问题中，并以不同的时间尺度承载着物理上重要的运动。规则和奇异摄动技术的微妙组合可以通过检测连接这些流形的邻域的显著解族来详细描述这种影响。这样的解描述了物理上不同的运动之间的重复的、复杂的转变自然界中的许多进化过程作为微分方程组的解。这些解的集合可以被视作高维相空间中的表面，而演化系统不同状态之间的不规则转变表现为连接这些表面的混沌轨道。该方案涉及当解的表面由具有多个不同时间尺度的解组成时，在重要且频繁的情况下检测这样的转变。提出的数学技术可以用来研究分子振动、海洋动力学中的表面波模式、行星系统的引力相互作用和工程结构的混沌振荡等尚未解释的细节。

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.