The authors apply the method for computing normal forms of functional differential equations (FDEs) introduced by T. Faria and L. T. Magalhães [J. Differ. Equations 122, No. 2, 201–224 (1995; Zbl 0836.34069)] to systems of delay-differential equations (DDEs) with a single delay and a Takens-Bogdanov bifurcation (the original paper of Faria and Magalhães had a scalar DDE as an illustrating example).
A puzzling feature of the discussion of the Takens-Bogdanov bifurcation is that the normal form presented in the paper differs qualitatively from the normal form in textbooks, say, [Yu. A. Kuznetsov, Elements of applied bifurcation theory (Applied Mathematical Sciences 112. New York, NY: Springer) (2004; Zbl 1082.37002)]. The unfolding parameters do not unfold the saddle-node bifurcation which should be present near a generic Takens-Bogdanov bifurcation. This peculiarity also shows up in the original paper of Faria and Magalhães.