Some Studies On Non-Uniformly Hyperbolic Attractors and the N-Body Problem

第一段：本研究计划由两个不同的研究项目组成。一个是关于n体问题，另一个是关于非均匀双曲吸引子的动力学。在第一个项目中，我们建议完成对希尔区域的初步分析，以获得空间四体问题的积分流形的详细描述。我们进一步提出研究二元碰撞正则化后积分流形的拓扑结构。在第二个项目中，我们提出研究一类强耗散非一致双曲吸引子，它们可以看作是公理a螺线管的自然扩展。这个新的类包括Henon吸引子和耗散扭转映射作为特殊情况。我们的研究项目包括映射的导数增长，吸引子的全局几何，符号动力学，拓扑熵和遍历理论在SRB测量和相关衰减方向。第二段：本研究计划由两个不同的研究项目组成。一个是关于theN-body问题，另一个是关于一类非均匀双曲吸引子的动力学。n体问题是在天体力学的悠久历史中用来研究天体运动的数学模型。它具有十个守恒定律，这些定律对所涉及的天体形成的可能的物理构型以及这些构型在三维物理空间中的可能方向施加了限制。我们的第一个项目是对这些限制的研究。这是一个长期存在的数学问题，在人造卫星轨道设计中具有潜在的应用价值。我们的第二个项目的目的是了解一类混沌动力系统的相干部分的几何结构。这类动力系统是从紊流和流体力学的研究中产生的。这些系统所涉及的混沌现象并不像所谓的“均匀双曲”系统那样被很好地理解。我们打算提供一个数学分析，以帮助更好地理解这些数学模型中涉及的混沌现象及其相应的物理过程

Paragraph one:This research proposal consists of two distinct research projects. One is on the N-body problem and the other is on the dynamics of non-uniformly hyperbolic attractors. In the first project we propose to complete our preliminary analysis on the Hill's region to get a detailed description of the integral manifold of the spatial 4-body problem. We further propose to study the topological structure of the integral manifold "after regularization of binary collisions In the second project, we propose to study a class of strongly dissipative non-uniformly hyperbolic "attractors that can be regarded as a natural extension of the Axiom A solenoid.This new class includes the Henon attractor and dissipative twist maps as special cases. Our program of study includes derivative growth of "maps, global geometry of the attractor, symbolic dynamics, topological entropy and ergodic theory in the direction of SRB measure and "correlation decay. Paragraph two:This research proposal consists of two distinct research projects. One is on theN-body problem and the other is on the dynamics of a class of non-uniformly hyperbolic attractors. The N-body problem is a mathematical model that has been used in the long history of celestial mechanics to study the motion of celestial bodies. It possesses ten conservation laws which impose restrictions on the possible physical configurations formed by the celestial bodies involved and the possible orientations of these configurations in three-dimensional physical space. Our first project is a study on these restrictions. This is a long standing mathematical problem with potential applications on the orbit design of artificial satellite. The aim of our second project is to understand the geometric structure of the coherent part of a class of chaotic dynamical systems. This class of dynamical systems are arisen from the study of turbulence and fluid mechanics. Chaotic phenomenon involved in these systems are not as well understood as that of the so call "uniformly hyperbolic" systems. We intend to provide a mathematical analysis that would help to better understand chaotic phenomenon involved in these mathematical models and their corresponding physical processes