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.

预测动力系统的多稳态之间的噪声诱发的临界过渡在各个领域都至关重要。本文研究了具有理想状态，次理想状态和不良状态的三稳态模型，作为实际系统的原型。然后，发现了由高斯白噪声引起的从理想状态到次理想状态（CT1）和从次理想状态到次理想状态（CT2）的两个关键转变。新的结果表明，噪声诱发的CT1和CT2发生在相应确定性系统的分叉点之前，并且随着噪声强度的增加，这种现象变得更早。因此，进一步探讨了噪声诱发的CT1和CT2的一些先验标准。首先，从新的角度将最大的Lyapunov指数和Shannon熵引入到噪声诱导的CT1和CT2的预测中。发现与经典方差和滞后1的自相关相比，它们都更有效，并且在强烈波动下，与CT1和CT2相比，香农熵更健壮。此外，在不安全区域的与参数有关的盆地中，可能会量化出可能发生噪声引起的临界转变的分叉参数范围。所有这些结果可能会为将来建立多个由噪声引起的关键转变的更一般的前体标准提供一些指导。发现与经典方差和滞后1的自相关相比，它们都更有效，并且在强烈波动下，与CT1和CT2相比，香农熵更健壮。此外，在不安全区域的与参数有关的盆地中，可能会量化出可能发生噪声引起的临界转变的分叉参数范围。所有这些结果可能会为将来建立多个由噪声引起的关键转变的更一般的前体标准提供一些指导。发现与经典方差和滞后1的自相关相比，它们都更有效，并且在强烈波动下，与CT1和CT2相比，香农熵更健壮。此外，在不安全区域的与参数有关的盆地中，可能会量化出可能发生噪声引起的临界转变的分叉参数范围。所有这些结果可能会为将来建立多个由噪声引起的关键转变的更一般的前体标准提供一些指导。