Dynamics and Numerical Analysis of State Dependent Delay Differential Equations

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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.