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A new technique for calculating the normal forms associated with the maprestricted to the center manifold of a class of parameterized maps near thefixed point is given first. Then we show the Takens-Bogdanov point of delaydifferential equations is inherited by the forward Euler method without anyshift and turns into a 1:1 resonance point. The normal form near the 1:1resonance point for the numerical discretization is calculated next by applyingthe new technique to the map defined by the forward Euler method. The localdynamical behaviors are analyzed in detail through the normal form. It showsthe Hopf point branch and the homoclinic branch emanating from theTakens-Bogdanov point are $O(\varepsilon)$ shifted by the forward Euler method,where $\varepsilon$ is step size. At last, a numerical experiment is carried toshow the results.

Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization