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In this work, we develop a homotopy continuation method to find the characteristic roots of delay differential equations with multiple delays. We introduce a homotopy parameter μ into the characteristic equation in such a way that for μ = 0 this equation contains only one exponential term (corresponding to the largest delay) and for μ = 1 the original characteristic equation is recovered. For μ = 0, all the characteristic roots can be expressed in terms of the Lambert W function. Pseudo-arclength continuation is then used to trace the roots as a function of μ. We demonstrate the method on several test cases. Cases where it may fail are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous. [ABSTRACT FROM AUTHOR]

This paper is a review of applications of delay differential equations to different areas of engineering science. Starting with a general overview of delay models, we present some recent results on the use of retarded, advanced and neutral delay differential equations. An emerging area for modeling with the help of delay equations is real-time dynamic substructuring, or hybrid testing. We introduce the main idea of this technique together with the latest advances. Special emphasis is given to the development of the theory and applications of partial delay differential equations. The review concludes with a summary of some open problems and questions concerning the analysis of spatially extended delayed systems. [ABSTRACT FROM AUTHOR]