Prolonged chaotic transients for parametric noise-induced critical transitions: Formation mechanism and scaling law

The underlying processes governing dynamical systems could create very long transient dynamics. In particular, in ecological systems, it is often difficult to distinguish this transient behavior from similar persistent dynamics. Chaotic transient is a typical class of long transients. However, even in low-dimensional dynamical systems, the formation mechanism and timescale of chaotic transients have not yet been fully understood. In this study, we establish a resilience metric for bounded parametric noise perturbations. It is shown that the parametric noise below the crisis point can induce chaos-related critical transitions. Chaotic transients for parametric noise-induced critical transitions are shown to be prolonged in the sense that the transient occurs on a timescale much longer than that of the regular chaotic transient. We further reveal the formation mechanism of prolonged chaotic transients and the transition pathway of escaped trajectories. The most striking fact is that in the vicinity of the critical noise, the average lifetime of chaotic transients follows a new scaling law as exp{α[ln(δ-δ_{c})]^{2}}, where δ_{c} is the noise threshold leading to the critical transition, and the α can be estimated by the reciprocal of the critical exponent of the corresponding deterministic crisis. We demonstrate the applicability of this scaling law using several classic maps and explore its applicable boundary.