Qualitative Analysis and Applications for Dynamical Systems with Time Delay

具有时滞和非线性的动力系统在真实的世界中很自然地出现，如信号传播、物种间相互作用、微生物学、流行病学、生理学等，但对它们的分析还没有引起足够的重视，特别是当时滞具有一定的分布且系统是对称的时候。在应用中，DS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进行完整的分类是至关重要的。虽然对于常时延DS系统已经有了一些理论和应用结果，但对于分布时延DS系统的理论分析还比较缺乏。此外，如果系统是对称的，动力学性质变得更加有趣。因此，有必要发展和推广延迟微分方程的理论和方法，使其得到更广泛、更深入的研究。在应用方面，本研究将集中在植物形态发生和互联网拥塞控制中的模式形成动力学，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。 这项研究将增加我们对具有时滞和对称性的DS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社区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，而且还为解决复杂的DS提供了实用工具，如生物，机械系统和互联网/神经网络模型。这些理论成果将极大地促进应用程序解决真实的问题，因为许多应用科学家可能不熟悉DDE理论，但他们确实希望应用该理论和方法来研究自己的具体问题。预期的成果将提供一个激励的环境，培训和受益于优秀的学生在应用动力系统领域。

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.