Preservation Of The T-B Bifurcation Of Delay Differential Equations Under Euler Method

We have known that this numerical scheme has a 1:1 resonance(the T-B point of map), that is, the Euler method preserves the singularity of delay differential equations. In this paper we reduce the numerical scheme mentioned above to a planar map on the center manifold based on the center manifold reduction and normal form calculation, witch has been used for the delay differential equations. Meanwhile, we find the relations between the parameters of the reduced form and the Euler scheme, and then obtain the bifurcation structure of the Euler scheme by analyzing the normal form we have:there exist a Neimark-Sacker bifurca-tion(the Hopf bifurcation for maps) branch and a discrete homoclinic bifurcation branch emanating from the origin on its parameter plane, finally we prove that the Neimark-Sacker bifurcation and discrete homoclinic bifurcation are respectively the O(h) perturbation of the Hopf bifurcation and homoclinic bifurcation that for the initial equation, that is, the Euler method preserves the T-B bifurcation structure of delay differential equations.