Dynamics of Nonlinear Differential Equations

非线性微分方程动力学

• 批准号: 0500674
• 负责人: John Mallet-Paret
• 金额: --
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500674&HistoricalAwards=false
• 关键词: Dynamics; Nonlinear; Differential; Equations

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.

应用数学布朗大学的John Mallet-Paret分部我们建议研究各种类型的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组和差分方程组时。要考虑的系统包括来自常微分方程组和偏微分方程组、格点微分方程组(即空间离散系统)、微分时滞方程和极大值加算子的系统。这些问题包括平衡点的存在性及其稳定性，空间格局和空间混沌的自发形成，旅行锋面的存在和定性，以及自发时间振荡的出现等问题。除了已有的微分方程式和动力学系统的理论和数值技术外，拟议研究的很大一部分还涉及开发和实施新的工具和技术来研究这些系统。所使用的方法包括奇异摄动、不变流形、指数二分法和拓扑法。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方法。成功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或进化行为，因为它发生在广泛的科学领域，包括生物、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--自发形成的模式、自我维持的振荡、内部反馈的调节(通常具有时滞)--这些都可以用动力系统理论的一些基本工具来分析。预计这些研究所产生的数学进步将增加对动力系统抽象理论的了解，并提供对科学研究领域的洞察。