Numerical computation of derivatives in systems of delay differential equations

This article deals with initial value problem solutions in systems of delay differential equations and their derivatives with respect to parameters, where the parameters may occur in the initial value, the initial function, the right-hand-side function, and the delay. Sufficient conditions for differentiability are given, and an efficient and reliable method for the numerical computation is presented. Emphasis is laid on the treatment of problems with a discontinuity at the initial time, for which it is shown that jumps occur in the derivative at the propagated discontinuity times. An explicit expression for the size of the jumps in the derivative is given. Features are discussed of the implementation of COLSOL-DDE, an experimental solver for initial value problems in delay differential equations that also computes the derivatives of the solution. The performance of the developed method is demonstrated by a comparison to standard techniques for derivative approximation.