Topics in Nonlinear Functional Differential Equations

非线性函数微分方程主题

• 批准号: 0701171
• 负责人: Roger Nussbaum
• 金额: $ 12.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701171&HistoricalAwards=false
• 关键词: Topics; Nonlinear; Functional; Differential; Equations

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.

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在C诱导的部分顺序中订单推迟。要研究的差异延迟方程是（1）ax'（t）= f（x（x（t），x（t），x（t- r_1），x（t- r_2），...，...，x（t- r_n）），其中a0和r_j可能依赖于函数x或r_j的历史X或r_j（x） x（t）。 主题（b）中所述类型的特殊类图由“广义最大算子”提供，努斯鲍姆和J. Mallet-Paret已显示出与公式（1）密切相关的。 即使是非常特殊的方程式（1），仍然是terra隐身。 例如，可以提及方程式（2）ax'（t）= -x（t） - （k_1）x（k_1）x（t -r_1） - （k_2）x（t -r_2），其中a0，k_j 0，r_j：= max（0，a_j +（a_j +（c_j）（c_j）x（c_j）x（c_j）x（t）），a_j 0 and a_j 0和c_ equection（a_j equiention necountion necountion necountion necountion necountion necountion ne treque（a），为0。 （1）（存在，稳定性和唯一性），（b）奇异扰动，例如，等式的周期性解决方案的限制行为。 （1）作为A --- 0，（c）解决方案的规律性（实际分析性？Gevrey类？），（d）方程式（1）和（e）对锥形贴图图F：C --- c的研究，既与它们与差分 - 票房方程以及与其他应用有关的研究。 多种科学现象，尤其是在数学生物学中，但在光学（例如，半导体激光器）和控制理论中也是由具有多个和/或可变时间滞后的差分 - delay方程来建模的。 尽管我们的主要重点是理论上的重点，但我们认为，理解等式等重要模型方程式方面的进展。 （1）和等式。 （2）上面将提供有关应用程序中出现的方程式的见解。 在等式的数值研究中，基本理论框架的重要性已经显而易见。 （2）根据参数提出了多种动态行为。在通常具有许多参数的应用程序中的方程式中，对理论框架的需求更加急切。