Takens-Bogdanov Bifurctaions In Differential Equations With Two Delays

Takens-Bogdanov bifurcation, as a kind of important bifurcations, reveals the mechanics of the Takens-Bogdanov point induces the Hopf point branch and the homoclinic branch. It also indicates the existence of homoclinic orbits. The Takens-Bogdanov bifurcation of differential equations with one constant delay has been well studied by Xu and Huang [Homolonic orbits and Hopf bifurcations in delay differential systems with T-B singularity, Journal of differential equations 244(2008) 582-598].In our work we carry the former results on Takens-Bogdanov bifurcations for differential equations with one constant delay over to the case of two constant delays. Feasible algorithms for the determination of the Takens-Bogdanov singularity for differential equations with two delays are given first. Next, by following the method of Faria and Magalhaes [Normal forms for retarded functional differential equations and application to Takens-Bogdanov singularity, Journal of differential equations 122(1995) 201-224]techniques of center manifold reduction and normal form calculation are applied to reduce the differential equations with two constant delays into an ordinary differential equation on the centre manifold. At last a direct analysis to the ordinary differential equations on the centre manifold leads to the bifurcation structures of the original differential equations with two constant delays.