Global Dynamics of Delay Differential Systems Modelling Nonlinear Feedbacks in Spatiotemporally Varying Environments

时空变化环境中非线性反馈建模的时滞微分系统的全局动力学

• 批准号: RGPIN-2019-06698
• 负责人: Wu, Jianhong
• 金额: $ 4.23万
• 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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739057
• 关键词: Global; Dynamics; Delay; Differential; Systems

The comprehensive Program will develop novel mathematical technologies to investigate implication of feedback delay on long-term dynamical behaviours of nonlinear epidemiological systems, and on computational performance of neural networks for high dimensional data clustering.  The mathematical framework is the so-called delay differential equations where the change rate of the system's state variable depends on both current and historical status of the system. These equations are also called functional differential equations as their solutions generate infinite dimensional semiflows on appropriate functional spaces.  Our overall objectives include: to identify and formulate several classes of delay differential equations arising from important applications in ecology, epidemiology, neural networks and data clustering; to develop frameworks, methodologies and technical tools to describe all possible behaviours of solutions of these equations; and to provide insights into the mechanisms behind observed/predicted dynamical scenarios of the underlying systems. We will integrate three intertwined thematic areas into a novel, cohesive and interdisciplinary Program with multiple projects on specific fundamental research problems and important applications.  In one thematic area, we develop the general theory of the global dynamics of semiflows which are monotone with respect to high rank cones, and its applications to delay equations with non-monotone and/or bi-directionally delayed feedback. This thematic research shall make groundbreaking advance towards identifying a broad class of nonlinear epidemiological and neural systems for which the classical monotone dynamical systems theory cannot be applied, and characterizing the global dynamics of such systems. In another thematic area, we develop technologies that can be effectively used to examine the creation, termination and global continuation of oscillatory patterns in models relevant to vector-borne disease spread. This thematic research will also establish the first generation of models and analytic tools to understand the complexity of co-feeding transmission of tick-borne diseases, contributing to the risk prediction of tick-borne pathogen spread under climatic and environmental changes. This risk prediction tool is much needed to build an effective public health decision support system. The thematic research on data clustering takes a novel dynamical systems approach to construct a clustering machine to address a fundamental challenge to adaptively select similarity thresholds so the change of similarity thresholds is linked to dynamical bifurcation when a new input is on the boundary of the domains of attraction of local attractors. Our innovative approach of clustering and our foundational research of the computational performance of a constructed neural network architecture can potentially lead to a major breakthrough in both theory and applications of high dimensional data clustering.

该综合计划将开发新的数学技术，以研究反馈延迟对非线性流行病系统的长期动态行为的影响，以及对高维数据聚集的神经网络的计算性能的影响。数学框架是所谓的时滞微分方程，其中系统状态变量的变化率取决于系统的当前状态和历史状态。这些方程也被称为泛函微分方程，因为它们的解在适当的函数空间上产生了无限维的半流。我们的总体目标包括：识别和制定几类在生态学、流行病学、神经网络和数据集群中的重要应用所产生的延迟微分方程；开发框架、方法和技术工具来描述这些方程的解的所有可能的行为；以及对潜在系统的观察/预测的动态场景背后的机制提供洞察力。我们将把三个相互交织的主题领域整合成一个新的、有凝聚力的、跨学科的计划，在具体的基础研究问题和重要应用上有多个项目。在一个主题领域，我们发展了关于高阶锥的单调半流的全局动力学的一般理论，以及它在具有非单调和/或双向时滞反馈的时滞方程中的应用。这一专题研究将取得突破性进展，以确定无法应用经典单调动力系统理论的广泛类别的非线性流行病学和神经系统，并表征这些系统的全球动力学。在另一个主题领域，我们开发了可有效用于检查与病媒传播疾病传播相关的模型中振荡模式的创建、终止和全球延续的技术。这项专题研究还将建立第一代模型和分析工具，以了解蜱传疾病共喂传播的复杂性，有助于预测气候和环境变化下蜱传病原体的传播风险。这种风险预测工具对于建立有效的公共卫生决策支持系统是非常必要的。数据聚类的主题研究采用了一种新的动力系统方法来构建聚类机，以解决自适应地选择相似阈值的基本挑战，从而将当新输入位于局部吸引子的吸引域边界上时，相似阈值的变化与动态分叉联系起来。我们的创新聚类方法和我们对构建的神经网络结构的计算性能的基础研究可能会在高维数据聚类的理论和应用方面带来重大突破。