Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a hopf bifurcation

In this paper, a novel method named pseudo-oscillator analysis is developed for the local dynamics near a Hopf bifurcation of scalar nonlinear dynamical systems with time delays. For this purpose, a pseudo-oscillator that is slightly perturbed from an undamped oscillator is firstly constructed, its fundamental frequency is the same as the frequency at the bifurcation point, and the disturbance is associated with the original system. Next, the pseudo-power function, defined as the power function of the pseudo-oscillator, is estimated along a harmonic function. Then we conclude that the local dynamics near the Hopf bifurcation can be justified from the variation of the averaged pseudo-power function. The new method features a clear physical intuition and easy computation, and it yields very accurate prediction for the periodic solution resulted from the Hopf bifurcation, as shown in three illustrative examples.