Remarks on the perturbation methods in solving the second-order delay differential equations

The paper presents a study on the validity of perturbation methods, such as the method of multiple scales, the Lindstedt-Poincaré method and so on, in seeking for the periodic motions of the delayed dynamic systems through an example of a Duffing oscillator with delayed velocity feedback. An important observation in the paper is that the method of multiple scales, which has been widely used in nonlinear dynamics, works only for the approximate solutions of the first two orders, and gives rise to a paradox for the third-order approximate solutions of delay differential equations. The same problem appears when the Lindstedt-Poincaré method is implemented to find the third-order approximation of periodic solutions for delay differential equations, though it is effective in seeking for any order approximation of periodic solutions for nonlinear ordinary differential equations. A possible explanation to the paradox is given by the results obtained by using the method of harmonic balance. The paper also indicates that these perturbation methods, despite of some shortcomings, are still effective in analyzing the dynamics of a delayed dynamic system since the approximate solutions of the first two orders already enable one to gain an insight into the primary dynamics of the system.