Dynamics and Numerical Analysis of State Dependent Delay Differential Equations

状态相关时滞微分方程的动力学和数值分析

• 批准号: 261389-2013
• 负责人: Humphries, Antony
• 金额: $ 1.09万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635624
• 关键词: Dynamics; Numerical; Analysis; State; Dependent

Many processes are modelled by differential equations subject to delays. In most models and mathematical theory this time delay is fixed, but in application areas there is much evidence of variable delays which depend on the state of the system. For example, the human hematopoietic system matures white blood cells faster when when mature cell counts are low. However, state dependency is often suppressed when deriving mathematical models of such processes, because of the paucity of the mathematical theory for state-dependent delay differential equations. Extension of the well-established theory and techniques for fixed delays to the state-dependent realm is currently receiving much attention, and the research program detailed here forms a part of this effort. In my research I will tackle state-dependent delay differential equations (DDEs) from three perspectives. Firstly I will study a model problem whose only nonlinearity is the state-dependency of the delays to understand the dynamics that state-dependency alone can drive. This will involve studying dynamics of and on invariant tori in state-dependent DDEs. Secondly, I will study numerical analysis of state-dependent DDEs, including derivation and implementation of new continuous Runge-Kutta methods and stability issues for state-dependent DDEs, as well as numerical techniques relevant to the invariant torus example. Thirdly I will consider state-dependent DDEs arising in applications including a human hematopoietic system model and Wheeler-Feynman electrodynamics. The techniques derived in the model problem and the numerical analysis will be used to further understanding of the application models, but it is also expected that applications will give rise to new mathematical questions.

许多过程都是由受延迟的微分方程建模的。在大多数模型和数学理论中，此时间延迟是固定的，但是在应用领域，有很多证据表明可变延迟取决于系统状态。例如，当成熟细胞计数低时，人类造血系统会更快地将白细胞成熟。但是，由于对状态依赖性延迟微分方程的数学理论的匮乏，在得出此类过程的数学模型时通常会抑制状态依赖性。目前，固定延误的固定延迟的理论和技术的扩展目前正在引起广泛关注，此处详细介绍的研究计划构成了这项工作的一部分。在我的研究中，我将从三个角度来解决国家依赖性延迟差分方程（DDE）。首先，我将研究一个模型问题，其唯一的非线性是延迟的状态依赖性，以了解单独依赖状态可以驱动的动态。这将涉及研究状态依赖性DDE中不变的Tori的动力学。其次，我将研究对状态依赖性DDE的数值分析，包括针对状态依赖性DDE的新连续runge-kutta方法的推导和实施，以及与不变型圆环相关的数值技术。第三，我将考虑在包括人类造血系统模型和Wheeler-Feynman电动力学在内的应用中产生的状态依赖性DDE。在模型问题和数值分析中得出的技术将用于进一步了解应用程序模型，但还可以预期应用程序将引起新的数学问题。