Delay effect on the relaxation oscillations of a van der pol oscillator with delayed feedback

This paper investigates the relaxation oscillations ofthe van der Pol oscillator with delayed feedback. Firstly, the stability ofslow manifold is determined and the bifurcation analysis on the fastsubsystem is carried out by means of stability switches. It is shown thatthe structure of the slow manifold changes essentially when the delayexceeds some critical value. Secondly, on the basis of geometric singularperturbation theory, it is shown that a small amplitude oscillation occursin the slow process of the relaxation oscillations due to the delay-inducedHopf bifurcation, and the period of the relaxation oscillations is shortenedas the delay increases. As demonstrated in the numerical simulations,oscillation with small amplitude in the slow process can be resulted frominner layer or Hopf bifurcation, or the both.