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

点击翻译按钮获取中文摘要

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.