Dynamics and Numerical Analysis of State Dependent Delay Differential Equations

许多过程都是用带有时滞的微分方程组来建模的。在大多数模型和数学理论中，这一时滞是固定的，但在应用领域中，有许多证据表明可变时滞取决于系统的状态。例如，当成熟细胞数量较少时，人类造血系统会更快地成熟白细胞。然而，由于状态相关时滞微分方程数学理论的匮乏，在推导这类过程的数学模型时，往往会抑制状态相关性。将固定延迟的成熟理论和技术扩展到依赖状态的领域目前正受到许多关注，这里详细说明的研究计划构成了这一努力的一部分。 在我的研究中，我将从三个角度处理状态依赖的延迟微分方程(DDE)。首先，我将研究一个模型问题，其唯一的非线性是时滞的状态依赖，以了解仅状态依赖可以驱动的动力学。这将涉及到研究状态依赖的DDES中不变环面的动力学。其次，我将研究状态依赖偏微分方程组的数值分析，包括新的连续Runge-Kutta方法的推导和实现，状态依赖偏微分方程组的稳定性问题，以及与不变环面例子相关的数值技巧。第三，我将考虑在应用中出现的状态依赖的偏微分方程组，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出的技术将被用来进一步理解应用模型，但也预计应用将产生新的数学问题。

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.