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颖的数学技术，以研究反馈延迟对非线性流行病学系统的长期动态行为以及高维数据聚类神经网络的计算性能的影响。​数学框架是所谓的延迟微分方程。其中系统状态变量的变化率取决于系统的当前和历史状态，这些方程也称为函数微分方程，因为它们的解在适当的函数空间上生成无限维半流。制定由生态学、流行病学、神经网络和数据聚类中的重要应用产生的几类延迟微分方程；开发框架、方法和技术工具来描述这些方程解的所有可能行为，并提供对观察到的机制的见解； /预测底层系统的动态场景。我们将把三个相互交织的主题领域整合到一个新颖的、有凝聚力的跨学科计划中，并在一个主题领域中开发针对特定基础研究问题和重要应用的多个项目。关于高阶锥体单调的半流全局动力学的一般理论，及其在具有非单调和/或双向延迟反馈的延迟方程中的应用本专题研究将在识别广泛的类别方面取得突破性进展。在另一个主题领域，我们开发了可有效用于检查创建、终止和全局的技术。这项专题研究还将建立第一代模型和分析工具，以了解蜱传疾病共同喂养传播的复杂性，从而有助于蜱传疾病的风险预测。气候和环境变化下传播的病原体传播非常需要这种风险预测工具来建立有效的公共卫生决策支持系统。数据聚类的专题研究采用了一种新颖的动态系统方法来构建聚类机器来解决这一基本挑战。自适应地选择相似性阈值，以便当新输入位于局部吸引子吸引域的边界时，相似性阈值的变化与动态分叉相关。我们的创新聚类方法以及我们对构建的神经网络的计算性能的基础研究。架构有可能导致高维数据聚类理论和应用的重大突破。