Topics in Nonlinear Functional Differential Equations and the Computation of Hausdorff Dimension

非线性泛函微分方程与Hausdorff维数计算专题

• 批准号: 1201328
• 负责人: Roger Nussbaum
• 金额: $ 15.3万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201328&HistoricalAwards=false
• 关键词: Topics; Nonlinear; Functional; Differential; Equations

This project proposes research concerning (a) the computation of the Hausdorff dimension of the invariant set for conformal, graph-directed iterated function systems, (b) Floquet multipliers for nonautonomous, linear functional differential equations and (c) smoothness questions (infinite differentiability versus real analyticity) for solutions of functional differential equations. Topic (a) is only loosely connected with topics (b) and (c); but a unifying theme in the study of all topics has been the use of the theory of positive operators and of generalizations of the Krein-Rutman theorem. For example, to an iterated function system as in (a), one associates a family of positive linear operators parametrized by positive real real numbers; and one proves that each such positive linear operator has a strictly positive eigenvector with a corresponding eigenvalue equal to the operator's spectral radius. The desired Hausdorff dimension is the unique value of the parameter for which the spectral radius of the operator equals one, and finding the Hausdorff dimension is facilitated by knowledge about the regularity of the eigenvectors. In studying topic (c) one finds examples of linear, nonautonomous functional differential equations which naively appear to be real analytic. Nevertheless, such equations may have infinitely differentiable periodic solutions which fail to be real analytic at an uncountable set of points. The theory of positive operators is helpful in constructing such examples.Many real-world problems are best modeled by equations which take history into account, so-called "functional differential equations" (FDE's) or "differential-delay equations." Thus red blood cell production in the human body at a given time involves knowledge of red blood cell levels six to ten days before and has been modeled by FDE's. Modeling the pupil light reflex in the human eye again involves FDE's. Many other examples can be found, not only in biology and physiology, but also in population dynamics, mechanical engineering (mechanical vibrations), nonlinear optics (lasers with opto-electronic feedback), economics and physics ( the famous and little-understood two body problem of electrodynamics). It is likely that advances in the theoretical understanding of FDE's will eventually have real-world applications. The search for such theoretical advances is a major focus of this proposal.

该项目提出了有关（a）的研究，以计算用于编制的，图形指导的迭代功能系统的不变设置，（（b）用于非自主性，线性功能差异方程式和（C）平滑度问题（无限差异与真实分析性）的floquet乘数用于函数型差异化的解决方案。主题（a）仅与主题（b）和（c）的主题松散相关；但是，在所有主题的研究中，一个统一的主题是使用积极的运营商理论和克雷因·鲁特曼定理的概括。例如，对于（a）中的迭代功能系统，一个人将正线性运算符的家族与正面实际实数参数相关联；一个人证明，每个这样的正线性算子具有严格的正征矢量，其特征值等于操作员的光谱半径。所需的Hausdorff尺寸是参数的独特值，该参数的光谱半径等于ON，并且通过了解特征向量的规律性的知识来促进Hausdorff尺寸。 在研究主题（c）中，人们发现了天真的非自主函数微分方程的示例，这些方程似乎是真实的分析。然而，这样的方程可能具有无限的周期性解决方案，这些解决方案无法在一组无数的点上进行真正的分析。 积极运算符的理论有助于构建此类示例。许多现实世界中的问题最好由考虑历史的方程式建模，这些方程式，所谓的“功能微分方程”（FDE）（FDE）或“微分 - 播放方程式”。 因此，在给定时间，人体中的红细胞产生涉及六到十天的红细胞水平的知识，并由FDE建模。 在人眼中对学生的光反射进行建模再次涉及FDE。不仅在生物学和生理学中，而且在人群动态，机械工程（机械振动），非线性光学元件（具有光电反馈的激光器），经济学和物理学（著名且鲜为人知的两种电动力学问题）中，还可以找到许多其他例子。对FDE的理论理解的进步可能最终将具有现实世界的应用。 寻找这种理论进步是该提案的重点。