Some Studies on Non-Uniformly Hyperbolic Attractors and The N-Body Problem

摘要DMS - 0204725。本课题主要研究非均匀双曲奇异吸引子。我们提出应用我们先前表述和研究的一个抽象集合来严格证明非一致双曲奇异吸引子在某些常微分方程系统中的存在性，包括一个修正的van der Pol系统和某些经历hopf分岔的周期激发系统。我们还提出用解析方法证明，当一个给定的常微分方程系统的渐近稳定周期解被一般外力周期性激发时，自然会产生非一致双曲奇异吸引子。这些非均匀双曲奇异吸引子的性质包括与混沌相关的大多数标准数学概念：正李亚普诺夫指数、正熵、SRB测度、相关指数衰减、轨道的良好符号编码等。我们还提出了对空间四体问题的积分流形的几何结构的研究。本研究计划主要研究非均匀双曲奇异吸引子。一般来说，吸引子是系统最终演化到的一种状态。研究自然界系统的最重要的方法之一是首先对其建模，然后将其作为某些微分方程的解进行研究。我们建议应用我们以前开发的一个理论来证明一些微分方程组中一类奇异吸引子的存在，这些微分方程组来自不同的科学学科，包括湍流、流体力学和等离子体力学的研究。我们要研究的吸引态具有非常复杂的动力学性质和复杂的几何结构。它们在许多数值和实验室实验中被观察到，但它们的存在在过去很少得到数学上的证实。我们还提出研究空间四体问题的积分流形的几何结构。这是一个长期存在的数学问题，在人造天体的轨道设计中具有潜在的应用前景。

ABSTRACT DMS - 0204725. PI: Qiu-Dong WangThis research proposal is mainly on the subject of non-uniformlyhyperbolic strange attractors. We propose to apply an abstract setting we previously formulated and studied to rigorously provethe existence of non-uniformly hyperbolic strange attractorsin certain systems of ordinary differential equations, including a modifiedsystem of van der Pol and certain periodically excited systems experiencingHopf bifurcations. We also propose to analytically establish the fact thatnon-uniformly hyperbolic strange attractors naturally arise whenan asymptotically stable periodic solution of a given systemof ordinary differential equations is periodically excited by genericexternal forcing. These non-uniformly hyperbolic strange attractors have properties that include most of standard mathematical notions associatedwith chaos: positive Liapunov exponents, positive entropy, SRB measure,exponential decay of correlation, nice symbolic coding of orbits etc.We also propose to study the geometric structure of the integral manifoldof the spatial four-body problem.This research proposal is mainly on the subject of non-uniformly hyperbolic strange attractors. In general an attractor is a state to which a system will eventually evolve. One of the most important way of studying systems in nature is to first model then study them as solutions of certain differential equations. We propose to apply a theory we previously developed to prove the existence of a class of strange attractors in some systems of differential equations arisen from various scientific disciplines including in the studies of turbulence, fluid mechanics and plasma mechanics. The attracting states we propose to study have very complicated dynamical properties and sophisticated geometric structures. They were observed inmany numerical and laboratory experiments but their existence were rarely established mathematically in the past. We also propose to study the geometricstructure of the integral manifold of the spatial four-body problem. This is a long standing mathematical problem with potential applications on orbit designfor artificial celestial objects.