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

该项目提出的研究内容包括：(A)共形、图向迭代函数系统不变集的Hausdorff维数的计算；(B)非自治线性泛函微分方程解的Floquet乘子的计算；(C)泛函微分方程解的光滑性问题(无穷可微性与实解析性)。主题(A)只与主题(B)和(C)有松散的联系；但在所有主题的研究中，一个统一的主题是使用正算子理论和推广Krein-Rutman定理。例如，对于(A)中的迭代函数系统，我们将一族由正实数参数表示的正线性算子联系在一起，并且证明了每个这样的正线性算子都有一个严格的正特征向量，其对应的特征值等于算子的谱半径。期望的Hausdorff维是算子的谱半径等于1的参数的唯一值，通过了解特征向量的正则性可以方便地求出Hausdorff维。在研究主题(C)时，人们发现了一些线性的、非自治的泛函微分方程的例子，它们看起来天真地看起来是真正的解析的。然而，这样的方程可能有无限可微的周期解，而这些周期解在不可数的点集上不是真正解析的。正算符理论有助于构建这样的例子。许多现实世界的问题最好用考虑历史的方程来模拟，所谓的“泛函微分方程”(FDE)或“微分-延迟方程”。因此，人体在给定时间产生的红细胞涉及到六到十天前的红细胞水平知识，并已由FDE进行了建模。对人眼中的瞳孔光反射进行建模再次涉及FDE。还有许多其他的例子，不仅在生物学和生理学中，而且在种群动力学、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、鲜为人知的电动力学两体问题)中都可以找到。很有可能，在对FDE的理论理解上的进步最终将在现实世界中应用。寻求这样的理论进步是这项提议的一个主要焦点。

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.