Summary: The paper is devoted to the investigation of a triple-zero singularity and the generalized associated eigenspace. A reduced form for parameterized delay differential systems is obtained by using center manifold reduction and normal form calculation. The results given in the paper by E. Freire, E. Gamero, A. J. Rodríguez-Luis
and A. Algaba
[Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, No. 12, 2799–2820 (2002; Zbl 1043.37042
)] are employed to analyze the bifurcation behavior of the parameterized delay differential system with triple-zero singularity in detail and an example is presented to illustrate the results.