Dynamics of Nonlinear Differential Equations

我们建议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。特别地，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。将研究与表现空间模式的晶格系统平衡的存在性、渐近于这种模式的行波解、这种波的各向异性（晶体学）钉住以及相关稳定性性质有关的问题。我们还建议研究这种时滞系统（循环系统），并观察由奇异扰动结构产生的模式的出现。拟议研究的很大一部分涉及新工具和技术的开发和实施，以及将现有的动力系统技术扩展到新的情况。我们正在发展新的数学技术，以便分析作为模型出现在各个科学领域的微分方程。广泛的科学学科被代表，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-模式的自发形成，自我维持的前沿和振荡，内部反馈的调节（通常带有时间延迟）-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知识，并将提供洞察力。

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.