On the Codimension of the Singularity at the Origin for Networked Delay Systems

Continuous-time dynamical networks with delays have a wide range of application fields from biology, economics to physics, and engineering sciences. Usually, two types of delays can occur in the communication network: internal delays ( due to specific internal dynamics of a given node) and external delays ( related to the communication process, due to the information transmission and processing). Besides, such delays can be totally different from one node to another. It is worth mentioning that the problem becomes more and more complicated when a huge number of delays has to be taken into account. In the case of constant delays, this analysis relies much on the identification and the understanding of the spectral values behaviorwith respect to an appropriate set of parameterswhen crossing the imaginary axis. There are several approaches for identifying the imaginary crossing roots, though, to the best of the authors' knowledge, the bound of themultiplicity of such roots has not been deeply investigated so far. This chapter provides an answer for this question in the case of time-delay systems, where the corresponding quasi-polynomial function has non-spare polynomials and no coupling delays. Furthermore, we will also show the link between this multiplicity problem and Vandermonde matrices, and give the upper bound for the multiplicity of an eigenvalue at the origin for such a time-delay system modeling network dynamics in the presence of time-delay.