Topics in Nonlinear Functional Differential Equations

对于NSFDMS-0701171(Nussbaum)中的非线性泛函微分方程组，Nussbaum提出了两个相关领域的研究：(A)具有状态相关时滞的非线性微分时滞方程的大的动力学问题和(B)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，并且在C诱导的偏序中保持有序。所研究的微分时滞方程的形式是(1)ax‘(T)=f(x(T)，x(t-r1)，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r_j：=r_j(x(T))可以简单地是x(T)的函数。由Nussbaum和J.Mallet-Paret所证明的与方程(1)密切相关的“广义极大+算子”提供了上述主题(B)中所描述的一类特殊类型的映射。即使方程(1)的非常特殊的情况仍然是未知的。例如，可以提到方程(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)奇摄动，例如，方程周期解的极限行为。(1)作为a-0，(C)解的正则性(实解析性？Gevrey类？)，(D)方程的Poincare-Bendixson理论(1)和(E)保锥映射F：C-C的研究，它们与微分时滞方程和其他应用的关系。各种各样的科学现象，特别是在数学生物学中，也在光学(例如半导体激光器)和控制理论中，已经由具有多个和/或可变时滞的微分延迟方程来建模。虽然我们的主要关注点是理论，但我们相信，在理解像EQ这样的重要模型方程方面的进展。(1)和等式(2)以上内容将提供对应用程序中出现的方程的深入了解。在方程的数值研究中，基本理论框架的重要性已经显而易见。(2)根据参数的不同，提出了各种不同的动力学行为。在通常具有许多参数的应用方程式中，对理论框架的需求更加迫切。

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.