Continuous Complexity and Dynamics

持续的复杂性和动态性

• 批准号: 9988809
• 负责人: Michael Shub
• 金额: $ 5.94万
• 依托单位: IBM Thomas J Watson Research Center
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988809&HistoricalAwards=false
• 关键词: Continuous; Complexity; Dynamics

Abstract :The Principal Investigator is Michael Shub. Shub and co-workersPugh, Wilkinson and others have established the stable ergodicity of awide range of partially hyperbolic dynamical systems which preserve asmooth volume element, thus establishing the statistical robustness ofthese chaotic systems. The thrust of the work proposed is to extendthese results to include almost all partially hyperbolic systems and toeliminate the hypothesis of volume preservation. This is an ambitious goal,but the results obtained included the development of new tools, namelyjulienne quasi-conformality and julienne density point preservation of thestable holonomy maps, whose power has not been fully explored. A second theme is a study of the complexity of continuous problems centered around theanalysis and application of variants of Newton's method.Chaotic dynamical systems are pervasive in scientific, engineering andnumerical simulation applications. The proposed work would lay foundationsfor the statistical analysis of these systems, whose deterministic behavior isunpredictable due to the presence of sensitive dependence on initialconditions. Newton's method for solving systems of equations is one ofthe main algorithms of numerical analysis. A better understanding of itsproperties and those of variants can make the solution of problems ofscientific and engineering origin more efficient.

摘要：首席调查员是迈克尔·舒布。Shub和他的同事Pugh，Wilkinson等人建立了保持光滑体积元的大范围部分双曲动力系统的稳定遍历性，从而建立了这些混沌系统的统计稳健性。提出的工作的主旨是将这些结果推广到几乎所有的部分双曲型系统，并消除体积保持的假设。这是一个雄心勃勃的目标，但所取得的成果包括开发新的工具，即稳定完整映射的julienne拟一致和julienne密度点保持，其功能尚未得到充分探索。第二个主题是围绕牛顿方法变体的分析和应用来研究连续问题的复杂性。混沌动力系统在科学、工程和数值模拟应用中普遍存在。所提出的工作将为这些系统的统计分析奠定基础，由于存在对初始条件的敏感依赖，这些系统的确定性行为是不可预测的。求解方程组的牛顿方法是数值分析的主要算法之一。更好地了解它的性质和变种的性质可以使解决科学和工程起源的问题更有效率。