Homoclinic orbits and Hopf bifurcations in delay differential systems with T–B singularity

The paper carries the results on Takens–Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity, J. Differential Equations 122 (1995) 201–224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens–Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T–B singularity in detail and present an example to illustrate the results.