Dynamics of Nonlinear Differential Equations

Pi：John Mallet-Paret，Brown University DMS-0200178摘要：我们建议对各种类型的非线性动力系统的定性性质进行基础性研究，特别是当它们来自微分差分方程组时。要考虑的系统包括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)、微分时滞方程和锥面上的非线性映射。这些方程的平衡点的存在性和稳定性，空间格局和空间混沌的自发形成，以及旅行前锋的存在性和定性性质等问题将被研究。除了已有的微分方程式和动力系统的理论和数值技术外，这项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。这些技术涉及奇异摄动、不变流形、指数二分法和拓扑学方法(Morse分解、不动点定理、度理论)。我们正在开发新的数学方法来分析和理解作为模型不变的科学领域出现的微分方程和差分方程。非常广泛地说，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、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(通常带有时滞)--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和科学研究领域的认识和洞察。

PI: John Mallet-Paret, Brown UniversityDMS-0200178Abstract:We propose a fundamental study of qualitative properties of various classesof nonlinear dynamical systems, in particular as they arise from differentialand difference equations. Among the systems to be considered are those fromordinary and partial differential equations, systems of lattice differentialequations (that is, spatially discrete systems), differential delay equations,and nonlinear maps on cones. Issues such as existence of equilibria and theirstability, spontaneous formation of spatial patterns and spatial chaos, andexistence and qualitative properties of traveling fronts, will be studied forthese equations. In addition to established techniques (both theoreticaland numerical) from differential equations and dynamical systems, a significantportion of the proposed research involves the development and implementationof new tools and techniques with which to study these systems. Among thetechniques to be employed are those involving singular perturbations,invariant manifolds, exponential dichotomies, and topological methods (Morsedecompositions, fixed point theorems, degree theory).We are developing new mathematical techniques for analyzing and understandingdifferential equations and difference equations which arise as models invarious areas of science. Very broadly, these types of mathematical systemsmodel time-dependent or evolutionary behavior, as it occurs in a wide rangeof scientific areas, including biology, chemistry, electrical circuit theory,image processing, and material science. While this is a very broad scope ofinquiry, the specific problems to be studied exhibit features in common --spontaneous formation of patterns, self-sustained oscillations, regulation byinternal feedback (often with time delays) -- which can be analyzed with someof the basic tools of dynamical systems theory. It is expected the resultingmathematical advances arising from these studies will increase the knowledgeof and will provide insight into both the abstract theory of dynamicalsystems and the scientific areas of inquiry.