Renormalization Group Methods For Delayed Differential Equations

In the social and natural sciences, the power system can be described as the differential equations in which the state variables change with time. Delays in the control system are quite common, strictly speaking, is inevitable. Studies have shown that ignoring the amount of time delay to study the system even with a small delay, will get the wrong conclusions, and some dynamic phenomena taking into account the time delay can get the correct interpretation. Therefore, more system dynamics with time-delay is more widely appreciated. In recent years, studies of delay differential equation theory have been into the stage of rapid development. People are increasingly concerned about the analytical solution, numerical solution, and the stability of the delay differential equations etc. This paper studies the approximate solution of delay differential equations.Perturbation theory as a branch of modern applied mathematics, is an effective method for the approximate solution of the nonlinear problems. The most commonmethods are direct perturbation method and singular perturbation method, however, our known perturbation methods for approximate solutions of the delay differential equations have some limitations. With the renormalization group method developed in recent years. We can overcome some inherent problems of traditional singular perturbation methods with the renormalization group method. Thus the renormalization group method has been widely used. Accordingly, we propose to use the renormalization group approach for the study of this topic.This paper presents the application of the renormalization group methods to delayed differential equations. By analyzing the delayed Duffing equation and Matheiu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that renormalization group approach provides approximate solutions which are practically superior to those obtained by the conventionally perturbation methods.