Analysis on Takens-Bogdanov points for delay differential equations

The paper analyzes the Takens-Bogdanov points for 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 an enlarged system for it is produced. Based on the descriptions for the eigenspace associated with the double zero eigenvalue, we reduce the enlarged system to a finite dimensional algebraic equation. The quadratic Takens-Bogdanov point, together with the corresponding values of parameters, is proved to be a regular solution of the reduced enlarged system and then can be computed by the standard Newton iteration directly.