Delay differential equations: theory and applications of periodicity, multistability and global dynamics

时滞微分方程：周期性、多稳定性和全局动力学的理论和应用

• 批准号: 105588-2011
• 负责人: Wu, Jianhong
• 金额: $ 3.42万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668963
• 关键词: Delay; differential; equations; theory; applications

A delay differential equation (DDE) describes the evolution of a system, for which the change rate of the state variable depends on the system's current and historical status. The initial state must be specified on an (initial) interval and an appropriate phase space must be infinite dimensional. This infinite dimensional functional analytic framework was developed in the last century, and the qualitative study of DDEs has been among major sources of inspiration for advance in nonlinear analysis and infinite dimensional dynamical systems. The objective of this proposal is to describe the global dynamics (how local structures such as equilibria, periodic/quasi-periodic solutions are connected together to form the global attractor) for several classes of nonlinear systems of DDEs arising from important applications including spatial dynamics of migratory birds and avian influenza; transmission dynamics of vector-borne diseases such as West Nile virus and Lyme disease; and neural networks for pattern storage and recognition. This project is the core component of a more comprehensive research program to understand long-term behaviors of biological or epidemiological systems with highly heterogenous individual/community structures. This core component focuses on developing general mathematical theories, methodologies and techniques to explore the implication of structure heterogeneity including environment/climate change for the spatiotemporal patterns of population dynamics. The heterogeneity to be addressed involve both temporal and spatial variations, including seasonality, long range dispersal vs local spatial diffusion, physiological structures and maturation stages, and state-dependent delays in signal processing and feedbacks. This project is an important step towards the long-term objective to understand the joint effect of time delay and spatial diffusion on dynamics of nonlinear systems, and its implications for a wide range of issues in biology, epidemiology, neuroscience and data mining. This project will provide ample opportunities for thesis topics and postdoctoral research, and NSERC support will be critical to form a well structured project team to fully capitalize a network of complementary expertise.**

时滞微分方程（DDE）描述系统的演化，其中状态变量的变化率取决于系统的当前和历史状态。初始状态必须在（初始）间隔上指定，并且适当的相空间必须是无限维的。这种无限维泛函分析框架是在上个世纪开发的，DDE 的定性研究一直是非线性分析和无限维动力系统进步的主要灵感来源之一。该提案的目的是描述 DDE 的几类非线性系统的全局动力学（局部结构，如平衡、周期/准周期解如何连接在一起形成全局吸引子），这些系统产生于重要的应用，包括空间动力学候鸟和禽流感；西尼罗河病毒和莱姆病等媒介传播疾病的传播动态；以及用于模式存储和识别的神经网络。该项目是一个更全面的研究计划的核心组成部分，旨在了解具有高度异质性个人/社区结构的生物或流行病学系统的长期行为。该核心部分侧重于发展通用数学理论、方法和技术，以探索结构异质性（包括环境/气候变化）对种群动态时空模式的影响。要解决的异质性涉及时间和空间变化，包括季节性、远距离扩散与局部空间扩散、生理结构和成熟阶段，以及信号处理和反馈中的状态相关延迟。该项目是朝着了解时间延迟和空间扩散对非线性系统动力学的共同影响及其对生物学、流行病学、神经科学和数据挖掘等广泛问题的影响的长期目标迈出的重要一步。该项目将为论文主题和博士后研究提供充足的机会，NSERC 的支持对于组建结构良好的项目团队以充分利用互补的专业知识网络至关重要。**