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

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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