Grant: Discovery Grants Program - Individual

Dynamical systems(DSs) with time delay and nonlinearity arise quite naturally in the real world, such as signal propagation, interaction of species, microbiology, epidemiology, physiology, as well as many others, and yet not so much attention has been paid to their analysis, particularly when the delay has certain distribution and the system is symmetric. One of the main purposes in the analysis of DSs in applications is to determine the (long-term) asymptotic behavior of solutions. It is critical to have a full classification of the structure of the qualitative behavior involved in a DS. Although there are some theoretic and applicable results for the DSs with constant delay, there is a lack of theoretical analysis in the systems with distributed time delay. If in addition the system is symmetric, the dynamical properties become much more interesting. Thus, it is necessary to develop and extend the theory and methodology to study delay differential equations(DDEs) deeply and widely. For the applications, this research will focus on the dynamics related to pattern formation in plant morphogenesis and Internet congestion control which are very active and challenging research areas in mathematical biology and engineering. This proposed research will increase our understanding of the qualitative behavior in DSs incorporating with time delay and symmetry. The theories, methodologies and algorithms developed in the proposal will have very high potential impact on the nonlinear dynamics community, not only strengthen the foundation for theoretical development in a large class of DDEs, but also provide a practical tool for solving complex DSs such as those arising from biological, mechanical systems and Internet/neural network models. The theoretical results will greatly facilitate the applications in solving real problems since many applied scientists may not be familiar with DDE theory, however they do want to apply the theory and methods to study their own specific problems. The anticipated outcomes will provide a stimulating environment for training and benefit to excellent student in the field of applied dynamical systems.