Computing the codimension of the singularity at the origin for delay systems in the regular case: A vandermonde-based approach

The analysis of time-delay systems mainly relies on the identification and the understanding of the spectral values bifurcations when crossing the imaginary axis. One of the most important type of such singularities is when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one known as Bogdanov-Takens singularity. Moreover, in some circumstances the codimension of the zero spectral value exceeds the dimension of the delay-free system of differential equations. To the best of the authors' knowledge, the bound of such a multiplicity was not deeply investigated in the literature. This paper provides an answer to this question for time-delay systems with linear part characterized in the Laplace domain by a quasipolynomial function with non sparse polynomials and without coupling delays.