Embedding of Delay Equations into an Infinite-Dimensional ODE System

A technique is introduced to embed the solution of the delay differential equation y′(t)=F(t, y(t), y(Θ(t))), y(0)=y0, into an infinite ODE system. This embedding is a starting point to deduce theorems on asymptotic stability of y, as well as to the construction of two numerical methods. The first method operates on a "natural gird," which is generated by iterating the lag function Θ, and it is accompanied by estimates on its speed of convergence. The second method consists of an application of standard ODE solvers to a truncated embedding. Its main merit is that the special structure of the embedded ODE system can be exploited in a massively parallel implementation.