Dynamics of Nonlinear Differential Equations

我们提出研究各种非线性动力系统的基本定性性质，因为它们是由微分方程产生的，特别是，我们的重点是致力于格动力系统（即空间离散系统）和微分延迟方程。将研究与晶格系统平衡点的存在性有关的问题，这些平衡点是空间模式的，行波解是这种模式的渐近解，这种波的各向异性（晶体学）钉扎，以及相关的稳定性。我们还建议研究这样的系统与时间延迟（循环系统），并观察patternsresulting从奇异摄动结构的出现。一个重要部分的拟议研究涉及新的工具和技术的开发和实施，以及现有的技术的动力系统扩展到新的情况下，我们正在开发新的数学技术，以分析微分方程出现在各个科学领域的模型。 代表了广泛的科学学科，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-自发形成的模式，自我维持的前沿和振荡，通过内部反馈进行调节（通常有时间延迟）--这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以下方面的知识，并将提供洞察到动力系统的抽象理论，和调查的科学领域。

We propose studying fundamental qualitative properties of various classesof nonlinear dynamical systems, as they arise from differential equations.In particular, our emphasis is devoted to lattice dynamical systems (thatis, spatially discrete systems) and differential delay equations. Questionsrelated to the existence of equilibria of lattice systems which exhibitspatial patterns, traveling wave solutions which are asymptotic to suchpatterns, anisotropic (crystallographic) pinning of such waves, and relatedstability properties, will be studied. We also propose to study such systemswith time delays (cyclic systems), and to observe the emergence of patternsresulting from a singular perturbation structure. A significant portion ofthe proposed research involves the development and implementation of newtools and techniques, as well as the extension of existing techniques ofdynamical systems to new situations.We are developing new mathematical techniques in order to analyze differentialequations which arise as models in various areas of science. A wide range ofscientific disciplines is represented, including biology, chemistry, electricalcircuit theory, image processing, and material science. While this is a verybroad scope of inquiry, the specific problems to be studied exhibit featuresin common -- spontaneous formation of patterns, self-sustained fronts andoscillations, 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 thesestudies will increase the knowledge of, and will provide insight into both theabstract theory of dynamical systems, and the scientific areas of inquiry.