Precursor criteria for noise-induced critical transitions in multi-stable systems

Predicting noise-induced critical transitions between multi-stable states of a dynamical system is of uttermost importance in various fields. This paper investigates a tri-stable model with desirable, sub-desirable and undesirable states as a prototype class of real systems. Then, two critical transitions, from the desirable state to the sub-desirable one (CT1) and from the sub-desirable state to the undesirable one (CT2), induced by Gaussian white noise are uncovered. The new results show that the noise-induced CT1 and CT2 take place before the bifurcation point of the corresponding deterministic system and this phenomenon becomes earlier with increasing noise intensity. Therefore, some precursor criteria of the noise-induced CT1 and CT2 are further explored. Firstly, the largest Lyapunov exponent and the Shannon entropy are introduced into the prediction of the noise-induced CT1 and CT2 from a new perspective. It is found that both of them are more efficient compared to the classic variance and autocorrelation at-lag-1, and the Shannon entropy is more robust as compared to CT1 and CT2 under strong fluctuations. Moreover, a range of the bifurcation parameter, where noise-induced critical transitions may occur, is approximately quantified in the parameter-dependent basin of the unsafe regime. All of these results may provide some guidance for establishing more general precursor criteria of multiple noise-induced critical transitions in the future.