Qualitative Analysis and Applications for Dynamical Systems with Time Delay

具有时滞和非线性的动力系统在现实世界中十分常见，如信号传播、物种相互作用、微生物学、流行病学、生理学等许多其他领域，但对其分析的关注并不多，特别是当时滞具有一定分布且系统是对称的情况下。在实际应用中，分析决策系统的主要目的之一是确定解的（长期）渐近行为。对DS中涉及的定性行为的结构有一个完整的分类是至关重要的。虽然对于等时延系统已有一定的理论和应用结果，但是对于分布时延系统的理论分析还比较缺乏。如果系统是对称的，那么动力学性质就变得有趣得多。因此，有必要发展和扩展对延迟微分方程进行深入和广泛研究的理论和方法。在应用方面，本研究将重点关注与植物形态发生模式形成相关的动力学和互联网拥塞控制，这是数学生物学和工程学中非常活跃和具有挑战性的研究领域。

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.