Control of bifurcation-delay of slow passage effect by delayed self-feedback

The slow passage effect in a dynamical system generally induces a delay in bifurcation that imposes an uncertainty in the prediction of the dynamical behaviors around the bifurcation point. In this paper, we investigate the influence of linear time-delayed self-feedback on the slow passage through the delayed Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. We perform linear stability analysis to derive the Hopf bifurcation point and its stability as a function of self-feedback time delay. Interestingly, the bifurcation-delay associated with Hopf bifurcation behaves differently in two different edges. In the leading edge of the modulating signal, it decreases with increasing self-feedback delay, whereas in the trailing edge, it behaves in an opposite manner. We also show that the linear time-delayed self-feedback can reduce bifurcation-delay in pitchfork bifurcation. These results are illustrated numerically and corroborated experimentally. We also propose a mechanistic explanation of the observed behaviors. In addition, we show that our observations are robust in the presence of noise. We believe that this study of interplay of two time delays of different origins will shed light on the control of bifurcation-delay and improve our knowledge of time-delayed systems.