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

一阶吸引子与N体问题的一些研究

基本信息

  • 批准号:
    0505594
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2005
  • 资助国家:
    美国
  • 起止时间:
    2005-07-01 至 2008-06-30
  • 项目状态:
    已结题

项目摘要

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 astate of periodic oscillation. However, as revealed by the modern theory ofdynamical systems, the real world is far morecomplicated. In fact, a dissipative system may evolve to a state of chaos.Thesechaotic statesare 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 anddynamic structure of rank one attractors, i.e., finding the underlining order ofthese chaotic behavior. Second, we try to establish with mathematical rigorthat these ``strange attractors'' are observable and are frequently encounteredin different discipline of science and mathematics. In the modern theory of dynamical systems, it is well acknowledgedthat the most important and intriguing objects are homoclinictangles. We know that horseshoes occur inside a homoclinic tangle.They are, however, only a relatively small part that is measuretheoretically ignorable. We also know that, in general, homoclinictangles are extremely complicated, perhaps so complicated thatcomprehensive understandings are hopeless to acquire. This proposal is mainlyon the study of a specific nonuniformlyhyperbolic homoclinic tangle (rank one attractors).First we hope tooffer a comprehensive understanding of the dynamical structure ofhomoclinic tangles of this kind by using many of the importantmathematical tools (Symbolic dynamics, SRB measures and relatedstatistical properties for a few) developed in the past onehundred years after the initial findings of the tangles byPoincar\'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.
这一建议是关于耗散系统的长期演化,特别是这些系统最终演化到的最终状态。直觉上,人们会期望一个稳定的平衡状态,或周期振荡状态。然而,正如现代动力系统理论所揭示的那样,现实世界要复杂得多。事实上,耗散系统可能演化为混沌状态。这种混沌状态就是所谓的“奇异吸引子”。本文对奇异吸引子的一个子类——秩一吸引子进行了数学分析。首先,我们试图了解一级吸引子的几何和动态结构,即找到这些混沌行为的下划线顺序。其次,我们试图用数学的严谨性来建立这些“奇怪的吸引子”是可观察到的,并且经常在不同的科学和数学学科中遇到。在现代动力系统理论中,公认最重要和最有趣的对象是同斜缠结。我们知道马蹄铁出现在同斜缠结中。然而,它们只是相对较小的一部分,在理论上是可以测量忽略的。我们也知道,一般来说,同临床纠缠是极其复杂的,也许复杂到无法获得全面的理解。本文主要研究了一类特殊的非均匀双曲同斜缠结(一级吸引子)。首先,我们希望通过使用许多重要的数学工具(符号动力学、SRB测量和一些相关的统计性质),在庞加莱最初发现缠结后的一百年里发展起来的,对这类同斜缠结的动力学结构提供一个全面的理解。其次,我们打算表明,像马蹄铁一样,这种同斜缠结在现实世界中经常出现,这将我们引向应用。

项目成果

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Qiu-Dong Wang其他文献

Qiu-Dong Wang的其他文献

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{{ truncateString('Qiu-Dong Wang', 18)}}的其他基金

Return Maps in Extended Phase Space for Non-autonomously Perturbed Equations
返回非自主微扰方程扩展相空间中的映射
  • 批准号:
    0758661
  • 财政年份:
    2008
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
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
  • 财政年份:
    2002
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
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:
    0196035
  • 财政年份:
    2000
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences Postdoctoral Research Fellowships
数学科学博士后研究奖学金
  • 批准号:
    9627756
  • 财政年份:
    1996
  • 资助金额:
    --
  • 项目类别:
    Fellowship Award

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