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

非均匀双曲吸引子与N体问题的一些研究

• 批准号: 0204725
• 负责人: Qiu-Dong Wang
• 金额: $ 7.81万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204725&HistoricalAwards=false
• 关键词: Some; Studies; Non; Uniformly; Hyperbolic

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.

摘要DMS - 0204725。 PI：王秋东本研究课题主要研究非均匀双曲型奇异吸引子。我们提出了一个抽象的设置，我们以前制定和研究严格证明存在非一致双曲奇怪吸引子在某些系统的常微分方程，包括修改系统的货车der Pol和某些周期性激发系统的经验Hopf分支。我们还建议建立解析的事实，即非一致双曲型奇怪吸引子自然出现时，一个渐近稳定的周期解的一个给定的系统的常微分方程周期性激发genericexternal forcing。这些非一致双曲型奇异吸引子的性质包含了与混沌相关的大多数标准数学概念：正Liapunov指数、正熵、SRB测度、相关性指数衰减、轨道的良好符号编码等.我们还提出研究空间四体问题积分流形的几何结构.一般来说，吸引子是一个系统最终会演化到的状态。研究自然界中系统的最重要方法之一是首先建模，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理论来证明存在一类奇怪的吸引子在一些微分方程系统中出现的各种科学学科，包括在研究湍流，流体力学和等离子体力学。我们所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被数学地确定。我们还提出研究空间四体问题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。