Dynamics of Nonlinear Differential Equations

抽象马莱-帕雷特 马勒-帕雷特将研究各种非线性动力系统的基本定性性质，因为它们来自微分方程。在系统中要考虑的是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，和非线性映射锥。 这些方程的问题，如平衡的存在性及其稳定性，图案的形成，空间混沌和行波，将被检查。 除了从微分方程和动力系统建立的技术（理论和数值），一个重要的部分，拟议的研究涉及开发和实施新的工具和技术来研究这些系统。其中的技术是采用涉及奇异摄动，不变流形，拓扑方法（莫尔斯分解，固定 点定理，度理论）。 马利特-帕雷特正在开发新的数学技术，用于分析和理解微分方程，这些微分方程在各个科学领域中作为模型出现。 代表了广泛的科学领域，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，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，通过内部反馈（通常具有时间延迟）进行调节-可以用动力系统理论的一些基本工具进行分析。 预计这些研究所产生的数学进步将增加知识，并将提供洞察力，动力系统的抽象理论和科学领域的调查。

Abstract Mallet-Paret Mallet-Paret will study fundamental qualitative properties of various classes of nonlinear dynamical systems, as they arise from differential equations. Among the systems to be considered are those from ordinary and partial differential equations, systems of lattice differential equations (that is, spatially discrete systems), differential delay equations, and nonlinear maps on cones. Issues such as existence of equilibria and their stability, pattern formation, spatial chaos, and traveling waves, will be examined for these equations. In addition to established techniques (both theoretical and numerical) from differential equations and dynamical systems, a significant portion of the proposed research involves the development and implementation of new tools and techniques with which to study these systems. Among the techniques to be employed are those involving singular perturbations, invariant manifolds, and topological methods (Morse decompositions, fixed point theorems, degree theory). Mallet-Paret is developing new mathematical techniques for analyzing and understanding differential equations which arise as models in various areas of science. A wide range of scientific areas is represented, including biology, chemistry, electrical circuit theory, image processing, and material science. While this is a very broad scope of inquiry, the specific problems to be studied exhibit features in common -- spontaneous formation of patterns, self-sustained oscillations, regulation by internal feedback (often with time delays) -- which can be analyzed with some of the basic tools of dynamical systems theory. It is expected that the resulting the mathematical advances arising from these studies will increase the knowledge of, and will provide insight into, both the abstract theory of dynamical systems, and the scientific areas of inquiry.