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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716258
• 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学系统的长期动力学行为的含义，以及对高维数据聚类的神经网络的计算性能。 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。这些方程式也称为功能微分方程，因为它们的解决方案在适当的功能空间上产生无限的尺寸半数。 我们的整体目标包括：识别和制定由生态学，流行病学，神经网络和数据聚类中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决方案的所有可能行为；并提供有关基础系统观察/预测动态场景背后的机制的见解。 我们将将三个相互交织的主题领域与有关特定基本研究问题和重要应用的多个项目集成到一个新颖，凝聚力和跨学科的计划中。 在一个主题领域，我们开发了半流量的全局动力学的一般理论，该理论相对于高等级锥是单调的，并且其应用于延迟非单调和/或双向延迟反馈的方程式。这项主题研究将使开创性的进步朝着确定一类广泛的非线性流行病学和神经系统，无法应用经典单调动力学系统理论，并表征这种系统的全球动态。在另一个主题领域，我们开发的技术可以有效地用于检查与媒介传播疾病扩散相关的模型中振荡模式的创造，终止和全球延续。这项主题研究还将建立第一代模型和分析工具，以了解tick传播疾病的共同进食的复杂性，从而有助于在气候和环境变化下tick传播病原体传播的风险预测。这种风险预测工具非常需要建立有效的公共卫生决策支持系统。对数据聚类的主题研究采用一种新型的动力学系统方法来构建聚类机器，以应对适应性选择相似性阈值的基本挑战，因此，当新输入在吸引人吸引人吸引人吸引人的领域边界上时，相似性阈值的变化与动态分叉有关。我们的聚类和我们对构建神经网络架构计算性能的基础研究的创新方法可能会导致理论和高维数据聚类的应用的重大突破。