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

本项目提出的研究涉及(a)保形、图向迭代函数系统不变集的Hausdorff维数的计算，(b)非自治、线性泛函微分方程的Floquet乘子，以及(c)泛函微分方程解的光滑性问题（无穷可微性与实解析性）。话题(a)与话题(b)、(c)仅有松散的联系；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克林-鲁特曼定理的推广。例如，对于(a)中的迭代函数系统，我们将一组正线性算子用正实数参数化；并且证明了每一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Hausdorff维数是算子的谱半径等于1的参数的唯一值，并且通过了解特征向量的正则性可以方便地找到Hausdorff维数。在研究主题(c)中，我们发现了一些线性的、非自治的泛函微分方程的例子，它们天真地表现为实解析方程。然而，这样的方程可能有无穷可微的周期解，在不可数点集合上不能成为实解析解。正算子理论有助于构造这类例子。许多现实世界的问题最好用考虑历史因素的方程来建模，即所谓的“泛函微分方程”（FDE）或“微分时滞方程”。因此，人体在特定时间内的红细胞生成涉及到6到10天前的红细胞水平，FDE已经建立了模型。模拟人眼的瞳孔光反射再次涉及到FDE。许多其他的例子可以找到，不仅在生物学和生理学，而且在人口动力学，机械工程（机械振动），非线性光学（激光与光电反馈），经济学和物理学（著名的和鲜为人知的电动力学二体问题）。对FDE的理论理解的进步很可能最终会有实际应用。寻找这样的理论进展是本提案的主要焦点。

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.