Chaotic ghosts in systems with parameter drift: Delay and control critical transitions

In dynamical systems with a time-dependent parameter, i.e., parameter drift, after crossing a saddle-node bifurcation, the so-called ghost state formed by the disappeared equilibria or periodic orbit can influence transient dynamics, causing a delayed transition. This phenomenon has been investigated previously. However, the effect of chaotic ghosts on the critical transition in drifting systems has been less studied. In this paper, we explore how chaotic ghosts and drifting rates influence critical transitions from the perspective of the ensemble. The results reveal the mechanism of the delayed transition related to chaos and how trajectories on the initial ensemble composed of a chaotic attractor transition to a qualitatively different object during the drift. In addition, we quantify the delayed transition and further find that the delay follows a power-law scaling with respect to the drifting rate. Finally, we show that the critical transition is fully avoided as long as the reversal rate of the parameter exceeds a certain critical rate, even though the bifurcation point has been crossed.