| Generally speaking,it is very hard to solve analytically the solutions to generic delay differential equations,hence a numerical method is required.A problem then comes naturally: whether the numerical method preserves the properties of the analytical solutions? Which deserves a further investigation.In 2014,Xu and Zou[1]have studied the preservation of Takens-Bogdanov bifurcation for delay differential equations under explicit Euler method,and showed that the numerical Hopf and homoclinic bifurcations are O(h)perturbation of their continous counterparts.In this paper,we mainly discuss the preservation of Takens-Bogdanov bifurcation for delay differential equations under linearθ-method.Firstly,we derive the parameterized map defined by the linear θ-method applied to the parameterized delay differential equations.Then we using the method by Xu and Zou to compute on the center manifold the normal form of this map,whose coefficients includes the bifurcation parameters.The normal form allows us to obtain the main numerical bifurcation structure for the original equation in the parameter plane :there exist a Neimark Sacker bifurcation curve and a numerical homoclinic bifurcation curve originating from the origin.Furthermore,comparing with the local representations of the bifurcations in the continuous case in the reference[2],we get the numerical Hopf bifurcation is O(h~2)perturbation of its continuous counterpart when θ =1/2 andQ = 2a,the numerical homoclinic bifurcation is O(h~2)perturbation of its continuous counterpart whenθ =1/2 and3Q = 5a,Q is the normal form’s coefficient of the mapping,otherwise,the numerical Hopf and homoclinic bifurcations are O(h)perturbation of their continuous counterparts.Particularly,we discussed the implicit Euler and the trapezoidal methods as the special cases of the linear θ-method.We lay our next effort to discuss the possibility that using our technique to prove the preservation of Takens-Bogdanov bifurcations of delay differential equations under Runge-Kutta method,or carry our the results to the case of multi-delay.Finally,we use numerical examples to illustrate our theoretical findings. |
