Topics in Nonlinear Functional Differential Equations

NSFDMS-0701171（Nussbaum）非线性泛函微分方程主题摘要 Nussbaum提出了两个相关领域的研究：（a）具有状态依赖时滞的非线性时滞微分方程的整体动力学;（B）关于映射F：C-C的问题，其中C是Banach空间中的闭锥，F是连续的、一次齐次的且在由C诱导的偏序中保序的。 所研究的微分时滞方程为（1）ax ′（t）=f（x（t），x（t-r_1），x（t-r_2），.，x（t-r_n）），其中a0和r_j可以取决于函数x的历史，或者r_j：=r_j（x（t））可以简单地是x（t）的函数。 在题目（B）中描述的一类特殊的映射由“广义最大-加算子”提供，Nussbaum和J. Mallet-Paret已经证明它与方程（1）密切相关。 甚至方程（1）的非常特殊的情况仍然是未知领域。 例如，可以提及等式（2）ax ′（t）= -x（t）-（k_1）x（t-r_1）-（k_2）x（t-r_2），其中a0，k_j 0，r_j =max（0，a_j +（c_j）x（t）），a_j 0和c_j不等于0。 将研究的领域包括（a）方程（1）的周期解（存在性、稳定性和唯一性），（B）奇异扰动，例如，方程周期解的极限行为(1)（c）解的正则性（真实的解析性？Gevrey类？），(d)方程（1）和（e）的Poincare-Bendixson理论，保锥映射F：C-C的研究，它们与微分延迟方程和其它应用的关系。 各种各样的科学现象，特别是在数学生物学，但也在光学（例如，半导体激光器）和控制理论已经通过具有多个和/或可变时滞的微分延迟方程来建模。 虽然我们的主要重点是理论，我们相信，在理解重要的模型方程，如方程的进展。(1)和等式(2)以上将提供对在应用中出现的方程的洞察。 一个基本的理论框架的重要性已经在EQ的数值研究中表现得很明显。(2)这表明取决于参数的各种各样的动力学行为。在应用程序的方程中，通常有许多参数，对理论框架的需求甚至更加迫切。

Abstract for NSFDMS-0701171 (Nussbaum) Topics in Nonlinear Functional Differential Equations Nussbaum proposes a study of two related areas: (a) the dynamics in the large of nonlinear differential-delay equations with state dependent time lagsand (b) questions about maps F:C---C, where C is a closed cone in a Banach space and F is continuous, homogeneous of degree one and order-preserving in the partial ordering induced by C. The differential-delay equations to be studied are of the form (1) ax'(t)=f(x(t), x(t- r_1), x(t- r_2), ...,x(t- r_n)), where a0 and r_j may depend on the history of the function x or r_j :=r_j (x(t)) may simply be a function of x(t). A special class of maps of the type described above in topic (b) is provided by "generalized max-plus operators", which Nussbaum and J. Mallet-Paret have shown are intimately related to equation (1). Even very special cases of equation (1) remain terra incognita. One may mention, for example, the equation (2) ax'(t)= -x(t) - (k_1)x(t -r_1) -(k_2)x(t -r_2), where a0, k_j 0, r_j :=max(0, a_j +(c_j)x(t)), a_j 0 and c_j is unequal to 0. Areas which will be investigated include (a) periodic solutions of equation (1)(existence, stability and uniqueness), (b) singular perturbations, e.g., limiting behaviour of periodic solutions of eq. (1) as a---0, (c) regularity of solutions (real analyticity? Gevrey class?), (d) Poincare-Bendixson theory for equation (1) and (e) studies of cone-preserving maps F:C---C, both in their relation to differential-delay equations and to other applications. A wide variety of of scientific phenomena, notably in mathematical biology, but also in optics (e.g., semiconductor lasers) and control theory, have been modeled by differential-delay equations with multiple and/or variable time lags. Although our main focus is theoretical, we believe that progress in understanding important model equations like eq. (1) and eq. (2) above will provide insight into equations which arise in applications. The importance of an underlying theoretical framework is already apparent in numerical studies of eq. (2) which suggest a wide variety of dynamical behaviour depending on the parameters. In equations from applications, which typically have many parameters, the need for a theoretical framework is even more acute.