Dynamics of Nonlinear Differential Equations

主要研究者：John Mallet-Paret，Brown UniversityDMS-0200178摘要：我们提出了一个基本的研究各种classesof非线性动力系统的定性性质，特别是因为它们来自微分和差分方程。 在系统中要考虑的是那些从普通和偏微分方程，系统的lattice微分方程（即，空间离散系统），微分延迟方程，和非线性映射锥。 这些方程的平衡点的存在性和稳定性，空间模式和空间混沌的自发形成，以及运动前沿的存在性和定性性质等问题将被研究。 除了从微分方程和动力系统建立的技术（理论和数值）之外，拟议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。其中的技术将采用涉及奇异摄动，不变流形，指数二分法，和拓扑方法（Morsedecompositions，不动点定理，度理论）。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出现在各个科学领域的模型。 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范围ofinquiry，具体的问题要研究展示的共同特点-自发形成的模式，自我维持的振荡，调节通过内部反馈（往往与时间延迟）-这可以分析与动力系统理论的一些基本工具。 预计从这些研究中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和科学领域的调查。

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.