Hopf Bifurcation Theory And Numerical Analysis In Some Types Of Delay Differential Equations

The system described by the delay differential equations not only depends on the present states, but also the past states, and it is because the existence of the time delay, what makes it more objectively in describing the actual problem, and more accurately in the solution of the problem. Delving into the dynamic characteristics of the delay differential system has important significance to understand these equation itself, and can promote the research of other field. In summary, studying the delay differential equations has important theoretical and practical value.Bifurcation is an important problem in the research of dynamic system theory, as an essential part of nonlinear dynamics, it is closely related to other dynamic behaviors, and generally as the basis to study. In order to understand the nonlinear phenomenon more clearly, the high codimensional bifurcation of the coupled system with delay has received widespread attention, such as codimension-2bifurcations. In this paper, we mainly study the Hopf bifurcation theory and numerical analysis in some types of delay differertial system, and detailed work is summarized as follows:Firstly, we study the existences of a class of codimension-2bifurcations and changes of the local topologies caused by bifurcations. And mainly involve the double Hopf bifurcation of the delay coupled dissipative Stuart-Landau oscillators. By discussing the eigenvalues of corresponding linear parts, the critical values of the double Hopf bifurcation is received. Using the center manifold method, we calculate the normal form when the system undergoes a non-resonant double Hopf bifurcation, find the bifurcation set near the equilibrium point, and give the complete classifications of the local topologies. The numerical simulations shows excellent agreement with the analytical predictions.Then, we research the numerical double Hopf bifurcation for delay coupled limit cycle oscillator, and find the necessary condition when the system undergoes a double Hopf bifurcation. By the theoretical inference, the critical conditions of the two parameters is given. The Euler method is used to discrete the original system, and the characteristic equation of the linear part of the discreted system is analyzed. The critical expressions of the numerical double Hopf bifurcation is obtained, then the maintainance of the numerical solution under the method of Euler is proved.Finally, in view of the van der Pol equation with two delays, the strict proof of numerical Hopf bifurcation under the Runge-Kutta method is given. Through the theoretical analysis, we ensure the equivalence of two delays, then one delay is fixed and another is chosen as the parameter, and the critical condition of the Hopf bifurcation is find. The system that discreted by the method of Runge-Kutta is analyse followed, and the result proves that the numerical solution can maintain the dynamic characteristics of the analytical solution, the simulation results are consistent with theoretical analysis.