Qualitative Analysis and Applications for Dynamical Systems with Time Delay

具有时滞和非线性的动力系统(DSS)在现实世界中非常自然地出现，如信号传播、物种相互作用、微生物学、流行病学、生理学等，但对其分析却没有得到太多的关注，特别是当时滞具有一定的分布且系统是对称的时。应用中决策支持系统分析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。虽然已有一些关于常时滞决策支持系统的理论和应用结果，但缺乏对分布时滞系统的理论分析。另外，如果系统是对称的，其动力学性质就会变得有趣得多。因此，有必要发展和推广研究时滞微分方程的理论和方法，使之更加深入和广泛。对于应用，本研究将集中于植物形态发生和网络拥塞控制中与模式形成相关的动力学研究，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。 这一研究将加深我们对决策支持系统中包含时滞和对称性的定性行为的理解。该提案中开发的理论、方法和算法将对非线性动力学领域产生非常高的潜在影响，不仅加强了一大类DDE的理论发展基础，而且为解决生物、机械系统和Internet/神经网络模型等复杂的DSS提供了实用工具。由于许多应用科学家可能不熟悉DDE理论，但他们希望将理论和方法应用于研究自己的具体问题，因此理论结果将极大地促进实际问题的应用。预期的结果将为应用动力系统领域的优秀学生提供一个激励的训练环境和受益。

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.