Multiple Scales Methods And Simulations For Periodic Solutions Of Two Kinds Of Delay Differential Equations

To reveal the essence of things,many complex mathematical models have been con-structed for the rapid developments of science and technology in various disciplines.Nonlinear problems have been research focus of science until now,and bifurcation is one of the im-portant nonlinear phenomenons.Since the twentieth century,delay differential models have been proposed in many disciplines,whose solution spaces are all infinite dimension.They are different with ordinary differential equation and their theoretical analysis is more dif-ficult.Therefore,the study of dynamics of time-delay differential equation is a challenging research field.Periodic solution is one of the important dynamics of time-delay system.The approximation of periodic solutions, investigation of their bifurcations of delay differential systems are very useful issues.In this paper we study two kinds of typical delay differential equations,One of them is delay Logistic equation, which is a kind of ecology model and has abundant dynamics.Another model is the damped oscillator with delayed feedback,which has more parameters than the first model and is a second-order differential equation.We approximate periodic solutions of these two models and analyze their stability by multiple scales analysis.One can use multiple scales method to do all those things,people need norm form to analyze stability for center manifold method. Until now,multiple scales analysis has been widely used in mechanic engineering.The results of this paper show that multiple scales analysis is effective for delay differential equation.To investigate period-doubling bifurcations of delay differential equations,we dis-cuss the existence of Feigenbaum constant,which is famous in the field of period-doubling bifurcation of discrete mappings.We explore the constant by simulation.