Numerical Analysis On Takens-Bogdanov Point In Delay Differential Equations

Based on the numerical calculation method for Takens-Bogdanov point in ordinary differential equations developed by Beyn W J[ Numerical analysis of homoclinic orbits emanating from a TB-point.IMA J Numer Anal,1994,14:381-460], in this paper, we investigate the numerical calculation method for Takens-Bogdanov point in delay differential systems with many constant delays and two parameters. By representing the delay differential equation as an abstract ordinary differential equation in its phase space, we produce an infinite dimensional algebraic system that defines the Takens-Bogdanov point. We then reduce the defining system into a finite dimensional algebraic equations which could be solved by standard algorithm for nonlinear equations. The system require to update itself during the iterative process. We proposed two transversality conditions as well,which guarantee that the Takens-Bogdanov point together with the corresponding critical parameter values are regular solutions of the reduced defining system, and consequently can be solved by Newton-like methods. At last, the proposed algorithm is implemented for several model problems, such as the delayed predator-prey system, neural network model and laser system model. The numerical results show that the method we proposed is effective for solving Takens-Bogdanov point in delay differential equations.