Dynamics of Nonlinear Differential Equations

非线性微分方程组（DMS-0500674） 约翰·马莱特-帕雷特 应用数学系 布朗大学我们建议研究各种非线性动力系统的基本定性性质，特别是当它们来自微分方程和差分方程时。要考虑的系统包括那些从常微分方程和偏微分方程，系统的格微分方程（即，空间离散系统），微分延迟方程，和最大加运营商。这些问题包括平衡点的存在性及其稳定性、空间模式和空间混沌的自发形成、运动前沿的存在性及其定性性质、自发时间振荡的出现等。除了从微分方程和动力系统中建立的技术（理论和数值）之外，所提出的研究的一个重要部分涉及研究这些系统的新工具和技术的开发和实施。在将要使用的技术中包括奇异摄动、不变流形、指数二分法和拓扑方法。我们正在发展分析和理解微分方程和差分方程的新的数学技术。这些方程通常作为模型出现在许多科学领域。从广义上讲，这些类型的数学系统模拟时间依赖或进化行为，因为它发生在广泛的科学领域，包括生物学，化学，电路理论，图像处理和材料科学。虽然这是一个非常广泛的调查范围，具体的问题要研究表现出共同的特点-自发形成的模式，自我维持的振荡，调节内部反馈（往往与时间延迟）-这可以被分析的一些基本工具的动力系统理论。预计这些研究所产生的数学进步将增加知识，并将提供洞察动力系统的抽象理论，以及科学领域的调查。

DYNAMICS OF NONLINEAR DIFFERENTIAL EQUATIONS (DMS-0500674) John Mallet-Paret Division of Applied Mathematics Brown UniversityWe propose to study fundamental qualitative properties ofvarious classes of nonlinear dynamical systems, in particular asthey arise from differential and difference equations. The systemsto be considered include those from ordinary and partial differentialequations, systems of lattice differential equations (that is,spatially discrete systems), differential delay equations, andmax-plus operators. Issues such as the existence of equilibria andtheir stability, spontaneous formation of spatial patterns and spatialchaos, the existence and qualitative properties of traveling fronts,and the appearance of spontaneous temporal oscillations will be studiedfor these problems. In addition to the established techniques (boththeoretical and numerical) from differential equations and dynamicalsystems, a significant portion of the proposed research involves thedevelopment and implementation of new tools and techniques with whichto study these systems. Among the techniques to be employed are thoseinvolving singular perturbations, invariant manifolds, exponentialdichotomies, and topological methods.We are developing new mathematical techniques for analyzing andunderstanding differential equations and difference equations. Suchequations typically arise as models in numerous areas of science. Inbroad terms, these types of mathematical systems model time-dependentor evolutionary behavior, as it occurs in a wide range of scientificareas, including biology, chemistry, electrical circuit theory, imageprocessing, and material science. Although this is a very broad scopeof inquiry, the specific problems to be studied exhibit features incommon -- spontaneous formation of patterns, self-sustained oscillations,regulation by internal feedback (often with time delays) -- which can beanalyzed with some of the basic tools of dynamical systems theory. It isexpected the resulting mathematical advances arising from these studieswill increase the knowledge of, and will provide insight into both theabstract theory of dynamical systems, as well as the scientific areas ofinquiry.