Mathematical Sciences: Dynamics in Almost Periodic Parabolic Equations and Coupled Map Lattices

小行星9402945 沈教授提出了两个研究领域：时间概周期抛物方程和格点动力系统。这两个领域在应用中经常遇到，例如，时间概周期抛物方程经常出现在神经脉冲传播理论和种群遗传学中，耦合映象格子作为描述流体动力学、化学反应系统、生物网络等各种时空结构的有用模型。许多重要的、根本性的问题还远没有得到很好的认识。在做时间概周期方程的研究，沉将专注于一般有界解的长期行为，特定的解决方案类型的存在性和它们的稳定性和分歧，全局吸引子的存在性和它们的结构。 同时，沉将特别注意那些属性的概周期方程不出现在周期的情况下（有强有力的证据，在她最近的工作和工作的许多其他研究人员的时间概周期方程是明显不同的周期）。在研究晶格动力系统时，沉计划开发一般方法来研究混沌解结构的存在性，并为这些类别的解建立稳定性准则和分叉场景。此外，沉打算给出一个详细的描述离散版本的某些微分方程的动力学行为有关的神经脉冲传播，流体动力学，化学反应系统等。 沈教授提出了两个研究领域：时间概周期抛物方程和格点动力系统。这两个领域在各个科学领域都经常遇到，如神经脉冲传播理论和群体遗传学，流体动力学中的混沌模式理论，化学反应系统，生物网络等。虽然这两个领域已经有了很多研究，但许多重要和基本的问题仍然没有得到很好的理解。在这项研究中，沈将专注于一般有界解的长期行为，特定解类型的存在性，以及时间概周期方程的全局吸引子的结构。此外，沉计划研究一般的晶格动力系统中的混沌解模式，并打算对与神经脉冲传播，流体动力学，化学反应系统等相关的某些微分方程的离散版本的动力学行为进行全面描述。

9402945 Shen Shen proposes research in two areas: time almost periodic parabolic equations and lattice dynamical systems. Both areas are frequently encountered in applications, for example, time almost periodic parabolic equations often occur in the theory of nerve pulse propagation and population genetics, coupled map lattices serve as useful models to describe various spatio-temporal structures in fluid dynamics, chemically reacting systems, biological networks, etc. Though much research has already been devoted to these two areas, many important and fundamental issues are still far from being well understood. In doing research on time almost periodic equations, Shen will focus on the long term behavior of general bounded solutions, the existence of specific solution types and their stability and bifurcations, the existence of global attractors and their structures. Meanwhile, Shen will pay particular attention to those properties of almost periodic equations which do not arise in the periodic case (there is strong evidence in her recent work and the work by many other researchers that time almost periodic equations are distinctively different from periodic ones). In doing research on lattice dynamical systems, Shen plans to develop general ways to study the existence of chaotic solution structures and to establish stability criteria and bifurcation scenarios for these classes of solutions. Moreover, Shen intends to give a detailed description of the dynamical behaviors for discrete versions of certain differential equations related to nerve pulse propagation, fluid dynamics, chemically reacting systems, etc. Shen proposes research in two areas: time almost periodic parabolic equations and lattice dynamical systems. Both areas are frequently encountered in various fields of sciences, such as the theory of nerve pulse propagation and population genetics, the theory of chaotic patterns in fluid dynamics, chemically reacting systems, biological networks, e tc. Though much research has already been devoted to these two areas, many important and fundamental issues are still far from being well understood. In this research, Shen will focus on the long term behavior of general bounded solutions, the existence of specific solution types, and the structure of global attractors for time almost periodic equations. Moreover, Shen plans to study chaotic solution patterns in general lattice dynamical systems and intends to give a comprehensive description of dynamical behaviors for discrete versions of certain differential equations related to nerve pulse propagation, fluid dynamics, chemically reacting systems, etc.