Homotopy Analysis for Periodic Motion of Time-Delayed Duffing System

Abstract

In this paper, the periodic motions of local dynamics of time-delayed oscillators near a single Hopf bifurcation have been investigated by means of the homotopy analysis method (HAM). With this technique, analytical approximations with high accuracy for all possible solutions are captured, which match the numerical solutions in the whole time regions. Two examples of dynamic systems are considered, which focus on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique lead to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales with strongly nonlinear examples. We studied the temporal dynamics of time-delayed systems in various regimes characterized by the parameters of the oscillator and the time delay parameter. The results given in this paper show that the time delay plays very important role in the analysis of multiply periodic motions with time-delayed systems. This paper is presented a general approach to the analysis of periodic motions of time-delayed systems. Although here we only consider a non-autonomous Duffing system with linear and nonlinear time-delayed position feedback, HAM can be extended to solve other time-delayed systems, such as coupled oscillators with time-delayed, feedback control which may have significance for the control of some physical or engineering systems.