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.

该项目提出的研究涉及 (a) 共形、图导向迭代函数系统的不变集的 Hausdorff 维数的计算，(b) 非自治、线性函数微分方程的 Floquet 乘子，以及 (c) 平滑问题（无限微分与实解析度）用于函数微分方程的解。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正算子理论和克赖因-鲁特曼定理的推广的使用。例如，对于（a）中的迭代函数系统，将一组由正实实数参数化的正线性算子关联起来；并且证明每个这样的正线性算子都有一个严格正的特征向量，其相应的特征值等于算子的谱半径。所需的豪斯多夫维数是算子谱半径等于 1 的参数的唯一值，并且通过了解特征向量的规律性可以方便地找到豪斯多夫维数。 在研究主题（c）时，人们会发现线性非自治函数微分方程的例子，这些方程天真地看起来是实解析的。然而，此类方程可能具有无限可微的周期解，这些解在不可数的点集上无法进行实解析。 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通过考虑历史的方程来建模，即所谓的“泛函微分方程”（FDE）或“微分延迟方程”。 因此，人体在给定时间的红细胞生成涉及六到十天前的红细胞水平知识，并且已通过 FDE 进行了建模。 人眼瞳孔光反射建模再次涉及 FDE。不仅在生物学和生理学中还可以找到许多其他例子，而且在群体动力学、机械工程（机械振动）、非线性光学（具有光电反馈的激光）、经济学和物理学（著名的和鲜为人知的两体）中也可以找到许多其他例子。电动力学问题）。 FDE 理论理解的进步很可能最终会在现实世界中得到应用。 寻求此类理论进展是本提案的主要焦点。