An Energy Analysis of the Local Dynamics of a Delayed Oscillator Near a Hopf Bifurcation

Hopf bifurcation exists commonly in time-delay systems. The local dynamics of delayed systems near a Hopf bifurcation is usually investigated by using the center manifold reduction that involves a great deal of tedious symbolic and numerical computation. In this paper, the delayed oscillator of concern is considered as a system slightly perturbed from an undamped oscillator, then as a combination of the averaging technique and the method of Lyapunov's function, the energy analysis concludes that the local dynamics near the Hopf bifurcation can be justified by the averaged power function of the oscillator. The computation is very simple but gives considerable accurate prediction of the local dynamics. As an illustrative example, the local dynamics of a delayed Lienard oscillator is investigated via the present method.