Differential Equations with State-Dependent Delays

Abstract

In this article an introduction to the wide field of retarded or delay differential equations with state-dependent delays is given. Hereby, the most important features of this type of differential equations which are profoundly different from ordinary differential equations are discussed. The presence of discontinuities in higher derivatives of the solution of delay differential equations require suitable integration methods. Thus, an efficient numerical code for the integration of delay differential equations is presented. This code is based on a fourth order RK one step algorithm. Furthermore, the presence of discontinuities may lead to severe problems when parameters are to be estimated using higher order optimization techniques. These problems are pointed out and a systematic way for the analysis of the smoothness of a least squares objective function is presented.

Stability and Hopf Bifurcation for a Cell Population Model with State-Dependent Delay

“…A Hopf bifurcation theorem for state-dependent delay differential equations has been recently obtained by Eichmann [26]. Numerical methods for solving state-dependent delay differential equations have been investigated by Hofer, Tibken, and Lehn [35], Shampine [56], and the references therein. Despite the difficulty related to the analysis of state-dependent delay differential equations, they have been used in some applied works-in cell biology, for instance, by Bélair [13], Bélair, Mackey, and Mahaffy [14], Hbid, Sanchez, and Bravo de la Parra [34], Hofer, Tibken, and Lehn [36], and Mahaffy, Bélair, and Mackey [49], but also in automatic [61].…”