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.

本项目提出的研究涉及（a）保形、图形定向迭代函数系统的不变集的Hausdorff维数的计算，（B）非自治、线性泛函微分方程的Floquet乘子和（c）泛函微分方程解的光滑性问题（无穷可微性与真实的解析性）。主题（a）与主题（B）和（c）仅有松散的联系;但所有主题研究中的一个统一主题是使用正算子理论和克莱因-鲁特曼定理的推广。例如，对于（a）中的迭代函数系统，我们将一个由正真实的真实的数参数化的正线性算子族联系起来;并且我们证明了每个这样的正线性算子都有一个严格正的特征向量，其对应的特征值等于算子的谱半径。期望的豪斯多夫维数是使算子的谱半径等于1的参数的唯一值，并且通过关于本征向量的正则性的知识来便于找到豪斯多夫维数。 在研究题目（c）时，我们会发现一些线性的、非自治的泛函微分方程，它们看起来是真实的解析的。然而，这样的方程可能有无穷可微的周期解，这些解在不可数的点集上不是真实的解析的。 正算子理论有助于构造这样的例子。许多现实世界的问题最好用考虑历史的方程来建模，即所谓的“泛函微分方程”（FDE's）或“微分延迟方程”。“因此，在给定时间内人体内的红细胞产生涉及六到十天前红细胞水平的知识，并且已经由FDE建模。 对人眼中的瞳孔光反射进行建模再次涉及FDE。许多其他的例子不仅可以在生物学和生理学中找到，还可以在人口动力学、机械工程（机械振动）、非线性光学（具有光电反馈的激光器）、经济学和物理学（著名但鲜为人知的电动力学两体问题）中找到。很可能FDE的理论理解的进步最终将有现实世界的应用。 寻求这种理论上的进步是这项建议的一个主要重点。