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

9402945沈申提出了两个方面的研究：时间概周期抛物方程和点阵动力系统。这两个领域在应用中都经常遇到，例如，神经脉冲传播理论和群体遗传学中经常出现时间几乎周期抛物方程，耦合图格是描述流体动力学、化学反应系统、生物网络等各种时空结构的有用模型。尽管对这两个领域已经进行了大量的研究，但许多重要和基本的问题仍然远远没有得到很好的理解。在时间概周期方程的研究中，沈将重点关注一般有界解的长期行为，特定解类型的存在性及其稳定性和分岔，全局吸引子的存在性及其结构。同时，沈将特别关注那些在周期情况下不会出现的概周期方程的性质（在她最近的工作和许多其他研究人员的工作中有强有力的证据表明时间概周期方程与周期方程有明显的不同）。在晶格动力系统的研究中，Shen计划开发研究混沌解结构存在性的一般方法，并为这类解建立稳定性判据和分岔场景。此外，Shen打算详细描述与神经脉冲传播，流体动力学，化学反应系统等有关的某些微分方程的离散版本的动力学行为。Shen建议在两个领域进行研究：时间概周期抛物方程和晶格动力系统。这两个领域经常出现在科学的各个领域，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。尽管对这两个领域已经进行了大量的研究，但许多重要和基本的问题仍然远远没有得到很好的理解。在这项研究中，沈将重点研究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，Shen计划研究一般晶格动力系统的混沌解模式，并打算对与神经脉冲传播、流体动力学、化学反应系统等有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.