Continuous Complexity and Dynamics

翻译后摘要：首席研究员是迈克尔·舒布。Shub和他的同事Pugh、威尔金森等人建立了一系列保持光滑体积元的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鲁棒性。所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方程组，并消除体积守恒的假设。这是一个雄心勃勃的目标，但所取得的成果包括新的工具，namelyjulienne准共形性和julienne密度点保存的稳定holonomy地图，其权力尚未得到充分的探讨。第二个主题是研究连续问题的复杂性，围绕牛顿方法的分析和应用。混沌动力系统在科学，工程和数值模拟应用中很普遍。这些系统的确定性行为由于对初始条件的敏感依赖性而不可预测，所提出的工作将为这些系统的统计分析奠定基础。求解方程组的牛顿法是数值分析的主要算法之一。更好地理解它的性质和变体的性质可以使科学和工程问题的解决更有效。

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.