Effects of the chaotic saddles on critical transitions in a piecewise linear oscillator subjected to parameter drift

Critical transitions are often used to describe the abrupt transitions between stable states when the control parameter crosses a bifurcation point. Alternatively, in chaotic systems, a suddenly disappeared chaotic attractor can transform into a saddle-type chaotic invariant set, called the chaotic saddle. Understanding how chaotic saddles influence critical transitions in drifting systems is crucial for predicting and controlling long-term dynamics. In this work we consider a piecewise linear system in which the driving amplitude varies linearly with time. The analysis shows that for both homoclinic and heteroclinic crises, when the time-dependent amplitude slowly crosses the crisis, the ensemble initially generated from the chaotic attractor remains around the chaotic saddle before suddenly collapsing to another coexisting attractor, resulting in a delay in the critical transition. In addition, we derive a power-law scaling relationship between the delayed transition and the drifting rate, where the scaling exponent depends on the pre-selected parameter interval containing chaotic saddles. Based on the delayed response of the drifting system, we reveal the necessary condition under which the collapse can be avoided, and successfully reverse the critical transition. Finally, we demonstrate that the crisis leading to the reappearance of the chaotic attractor can be masked or hidden due to the separation between the drifting rate and the escape rate.