Computing the Codimension of the Singularity at the Origin for Delay Systems: The Missing Link with Birkhoff Incidence Matrices
Résumé
A standard framework in analyzing Time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/geometric). This paper focuses on an interesting type of such singularities; that is when the zero spectral value is multiple. The simplest case,whichisquitecommoninapplications,ischaracterizedby an algebraic multiplicity two and a geometric multiplicity one known as Bogdanov-Takens singularity. Unlike finite dimen- sional systems, the algebraic multiplicity of the zero spectral value may exceed the dimension of the delay-free system of differential equations. To the best of the authors’ knowledge, the bound of such a multiplicity for Time-delay systems was not deeply investigated in the literature. Our contribution is two fold. First, we emphasize the link between the multiplicity characterization and Birkhoff matrices. Secondly, we elaborate a constructive bound for the zero spectral value in the regular case; i.e. when the delay polynomials of a given quasipolynomial are complete, as well as in the singular case; i.e. when such polynomials are sparse. In the last case, the established bound is sharper than Polya-Szego generic bound.
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