Numerical Computation of Takens-Bogdanov Points for Delay Differential Equations

The paper presents a numerical technique for computing directly the Takens-Bogdanov points in the nonlinear system of differential equations with one constant delay and two parameters. By representing the delay differential equations as abstract ordinary differential equations in their phase spaces, the quadratic Takens-Bogdanov point is defined and a defining system for it is produced. Based on the descriptions for the eigenspace associated with the double zero eigenvalue, we reduce the defining system to a finite dimensional algebraic equation. The quadratic Takens-Bogdanov point, together with the corresponding values of parameters, is proved to be the regular solution of the reduced defining system and then can be approximated by the standard Newton iteration directly.