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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571020
• 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.

许多过程都是由具有时滞的微分方程来描述的。在大多数模型和数学理论中，这个时间延迟是固定的，但在应用领域中，有很多证据表明可变延迟取决于系统的状态。例如，当成熟细胞计数低时，人类造血系统使白色血细胞更快地成熟。然而，状态依赖性往往是抑制推导数学模型时，这样的过程，因为缺乏的数学理论的状态依赖延迟微分方程。扩展完善的理论和技术，固定延迟的状态依赖领域目前正受到广泛关注，这里详细介绍的研究计划形成了这一努力的一部分。 在我的研究中，我将从三个角度来处理状态依赖延迟微分方程（DDE）。首先，我将研究一个模型问题，其唯一的非线性是状态依赖性的延迟，以了解状态依赖性单独可以驱动的动态。这将涉及到研究动态和不变环面的状态依赖DDE。其次，我将研究状态相关动态微分方程的数值分析，包括新的连续龙格库塔方法的推导和实现以及状态相关动态微分方程的稳定性问题，以及与不变环面示例相关的数值技术。第三，我将考虑状态依赖的DDE的应用，包括人类造血系统模型和惠勒-费曼电动力学。在模型问题和数值分析中得到的技术将用于进一步理解应用模型，但也可以预期，应用将产生新的数学问题。