Mathematical Sciences: Topics in Dynamical Systems

小行星9306265黑尔 这一建议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，集中在边界条件，扩散系数，耗散系数和域的形状上。 由这样的方程定义的流的比较将主要限于全局吸引子。 对于泛函微分方程，我们将主要对奇异摄动问题感兴趣，这些问题的解与由极限方程定义的映射的性质有关。 许多物理系统都是由定义在有界区域上的偏微分方程来建模的。 研究领域的影响（其形状等）是至关重要的 如果域的几何属性作为控制变量，例如，包括在最优形状设计的名称下的那些属性，以及其他属性，则尤其如此。 而且，许多域在某些方向上很小，并且为了设计以及计算目的，将域减少到较低维域是有利的。本提案涉及这类问题。 在激光光学以及生态学和生物学中的模型中，人们会遇到奇摄动延迟微分方程。 在这种情况下，奇摄动参数等于零的极限方程是映射。 在这个建议中，我们解决的延迟方程的动态和动态的地图之间的关系。 ***

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. ***