Mathematical Sciences: Topics in Dynamical Systems

数学科学：动力系统主题

• 批准号: 9306265
• 负责人: Jack Hale
• 金额: $ 12万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9306265&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Dynamical; Systems

9306265 Hale The thrust of this proposal is to continue to develop the theory of infinite dimensional dissipative systems by the consideration of specific types of problems. For example, in Partial Differential Equations, we study the dependence of the flows upon parameters, concentrating on the boundary conditions, diffusion coefficients, dissipation coefficients and the shape of the domain. Comparison of flows defined by such equations will be restricted mostly to the global attractors. For Functional Differential Equations, we will be primarily interested in singular perturbation problems which relate the solutions to the properties of maps defined by limit equations. Many physical systems are modeled by partial differential equations defined on a bounded domain. The study of the effects of the domain (its shape, etc.) on the dynamics is of paramount importance. This is especially true if the geometric properties ofl the domain act as a control variable, as, for example, those included under lthe name lof optimal shape design, as well as others. Also, many domains are small in some directions and it is advantageous to make a reduction to a lower dimensional domain for design as well as computational purposes. The present proposal addresses questions of this type. In lazer optics and models in ecology and biology, one encounters singularly perturbed delay differential equations. In such cases, the limit equation for the singularly perturbed parameter equal to zero is mapping. We address in this proposal the relationships between the dynamics of the delay equation and the dynamics of the map. ***

9306265 Hale 该提案的主旨是通过考虑特定类型的问题来继续发展无限维耗散系统理论。 例如，在偏微分方程中，我们研究流动对参数的依赖性，重点关注边界条件、扩散系数、耗散系数和域的形状。 由这些方程定义的流量的比较将主要限于全局吸引子。 对于泛函微分方程，我们主要对奇异摄动问题感兴趣，这些问题将解与极限方程定义的映射的属性联系起来。 许多物理系统都是通过在有界域上定义的偏微分方程来建模的。 研究域（其形状等）对动力学的影响至关重要。 如果域的几何属性充当控制变量，例如那些包含在最佳形状设计名称下的变量以及其他变量，则尤其如此。 此外，许多域在某些方向上都很小，并且为了设计和计算目的而减少到较低维度的域是有利的。本提案解决了此类问题。 在激光光学以及生态学和生物学模型中，人们会遇到奇扰动延迟微分方程。 在这种情况下，奇扰动参数为零的极限方程是映射。 我们在这个提案中解决了延迟方程的动力学和地图的动力学之间的关系。 ***