Numerical Computation Of T-B Point In Delay Differential Equations

In this work, we mainly study the numerical analysis of bifurcation theory, especiallythe numerical computation of Takens-Bogdanov point in delay differential equations. The work produces a robust method for the computation of Takens-Bogdanovpoint in delay differential equations, it is an original work, and enlarges the computational theory of bifurcation in delay differential equations. Besides, our method can also be applied in lots of areas such as engineering, ecology, mathematicalbiology, chemistry, medicine, economics, etc.In the dissertation, we consider a parameterized delay differential equation as followswhereλandμare two real parameters. The most interesting problem we cared in the dissertation is to compute the Takens-Bogdanov point in the equation.For this purpose, we first give a definition for quadratic Takens-Bogdanov point in the case of delay differential equation. For such a definition, we should show it is well-defined, that is to say, the definition should coincide with the case of ordinary differential equation because a delay differential equation can be viewed as an abstract ordinary differential equation, such a work is carried in section 2. Next, we give an enlarged system for computation of Takens-Bogdanov point in delay differential equation according to the description of Takens-Bogdanov singularity, and we proved such a system is regular at Takens-Bogdanov point in section 3. At last, we solve the enlarged system by Newton method in section 4.Besides, a numerical example is fulfilled in section 4. As a numerical example, we considered a delayed predator-prey systemwhere r,K,a,μ,D andτare positive constants. The numerical results shown that our method is an effective method for solving Takens-Bogdanov point in delay differentialequation.