Dynamics of Nonlinear Differential Equations

布朗大学应用数学系John Mallet-Paret我们建议研究各类非线性动力系统的基本定性性质，特别是由微分方程和差分方程产生的非线性动力系统。要考虑的系统包括常微分方程和偏微分方程系统、格微分方程系统（即空间离散系统）、微分延迟方程和max-plus算子。针对这些问题，将研究平衡的存在性及其稳定性、空间格局和空间混沌的自发形成、行进锋的存在性和性质以及自发时间振荡的出现等问题。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，拟议研究的很大一部分涉及开发和实施用于研究这些系统的新工具和技术。所采用的技术包括奇异摄动、不变流形、指数二分类和拓扑方法。我们正在开发新的数学技术来分析和理解微分方程和差分方程。这类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了依赖时间的进化行为，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研究范围，但要研究的具体问题表现出共同的特征-模式的自发形成，自我持续振荡，内部反馈调节（通常具有时间延迟）-可以用动力系统理论的一些基本工具进行分析。预期从这些研究中产生的数学进步将增加知识，并将提供对动力系统抽象理论以及科学探索领域的见解。

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.