Distinctive roles of hysteresis, amplitude death and oscillation death in generating fast-slow phenomena in parametrically and externally excited systems

Fast-slow phenomena are often referred to as time series showing clearly separated time scales of alternations between consecutive high-frequency large-amplitude oscillations and much smaller amplitudes or much more infrequent or quiescent intervals, which can be observed in many nonlinear systems. In this work, we show distinctive roles of hysteresis, amplitude death and oscillation death mechanisms in generating fast-slow switching behavior by using the paradigmatic models of Duffing and Van der Pol systems which are driven by both parametric and external excitations for the case when the frequencies of excitations are 1:1 and 1:2 rationally correlated. In the Duffing system, we show that hysteresis is the main mechanism to generate fast-slow behavior and the fast oscillations are resulted from the transient processes between different steady states. On the other hand, in the Van der Pol system, amplitude death and oscillation death are the fundamental mechanisms because of Hopf bifurcation. In this case, fast oscillations mainly correspond to the limit cycles, while slow quiescent intervals are due to the stable fixed points. In addition, we provide a rather simple model to implement responding lags of Van der Pol oscillators to the same external forcing, resulting in fast-slow switching relaying scenarios among subsystems. All theoretical results have been confirmed precisely by numerical simulations.