Delay Effects On The Takens-Bogdanov Singularities In Differential Equations

The paper lays efforts on the Takens-Bogdanov singularities in ordinary and delay differential equations. Based on the description for Takens-Bogdanov point in delay differential equations developed by Xu and Huang, [Homoclinic orbits and Hopf bifurcations in delay differential systems with Takens-Bogdanov singularity. J Differential Equations, 2008,244:582-598.], the sufficient and necessary conditions are given for that a differential equation with a Takens-Bogdanov singularity, when be added to some delay terms or removed the delay terms (by letting the delayτ=0), succeeds the Takens-Bogdanov bifurcation. It plays a very important instructional role in improving existed models in various scientific research fields, moreover, the works of computation and bifurcation analysis also can be reduced greatly. Besides, we present an example to illustrate the results.