The Preservation Of Newmark Method For Hopf Bifurcation Of Second-order Differential Equations

Properties of bifurcation for differential equation is widely applied to many fields, such as natural science and engineering technology. With the rapid speed development of computer, more and more scholars have devoted to establishing numerical methods which can correctly reflect and simulate properties of bifurcation for the original systems.In the field of structure dynamics and seismic calculation, applications of Newmark method depend on the dynamics of second order differential equations, especially in the stability of the equilibrium. With different forms of bifurcation phenomena, Hopf bifurcation is a kind of phenomenon with vital implications in industrial manufacture. The system is unstable when it undergoes a Hopf bifurcation in the neighborhood of the fixed point. So there is close relationship between Hopf bifurcation and stability of systems. Therefore, it is of great significance to consider properties of Hopf bifurcation for second order differential equations by using Newmark method.The preservation of Hopf bifurcation for second order differential equation with Newmark method is considered in this paper. The content is divided into three parts as follows.Firstly, the background of Newmark method and Hopf bifurcation is introduced, and so as in the research actuality of Hopf bifurcation for numerical discrete system both at home and abroad.Secondly, the equivalence of the property of Hopf bifurcation for second order differential equation and Neimark-Sacker bifurcation for numerical discrete system is investigated. The application of Newmark method obtains the existence of Neimark-Sacker bifurcation for numerical discrete system. Then, conversely, the existence of Hopf bifurcation for continuous system is proved when there exists Neimark-Sacker bifurcation. A numerical example of nonlinear second order differential equation is given to illustrate our results in the end.Thirdly, Hopf bifurcation for second order delay differential equation is existed, through choosing the delay as variable, we can testify that the numerical discrete system has Neimark-Sacker bifurcation. That is to say, the Hopf bifurcation for the original system can be preserved by Newmark method. Furthermore, we discuss the stability of the closed invariant curves. Then, numerical example of second order delay differential equation is chosen to demonstrate the feasibility of our main results.