Some Studies on the Rank One Attractors and the N-Body Problem

University Of Arizona, Tucson AZ

Investigators

Abstract

This proposal is about the long term evolution of systems with dissipation, especially the final states to which these systems evolve at the end. Intuitively, one would expect a state of stable equilibrium, or a state of periodic oscillation. However, as revealed by the modern theory of dynamical systems, the real world is far more complicated. In fact, a dissipative system may evolve to a state of chaos. Thesechaotic states are the so called ``strange attractors". This proposal is on a mathematical analysis of rank one attractors, a subclass of strange attractors. First we try to understand the geometric and dynamic structure of rank one attractors, i.e., finding the underlining order of these chaotic behavior. Second, we try to establish with mathematical rigor that these ``strange attractors'' are observable and are frequently encountered in different discipline of science and mathematics. In the modern theory of dynamical systems, it is well acknowledged that the most important and intriguing objects are homoclinic tangles. We know that horseshoes occur inside a homoclinic tangle. They are, however, only a relatively small part that is measure theoretically ignorable. We also know that, in general, homoclinic tangles are extremely complicated, perhaps so complicated that comprehensive understandings are hopeless to acquire. This proposal is mainly on the study of a specific nonuniformly hyperbolic homoclinic tangle (rank one attractors). First we hope to offer a comprehensive understanding of the dynamical structure of homoclinic tangles of this kind by using many of the important mathematical tools (Symbolic dynamics, SRB measures and related statistical properties for a few) developed in the past one hundred years after the initial findings of the tangles by Poincar\'e. Second we intend to show that, like the horseshoes, homoclinic tangles of this kind commonly occur in the real world, which lead us to applications.