Multiple tipping phenomena in a system with a time-dependent parameter excited by coloured noise

Tipping can describe various dynamical transitions in real-world systems, such as thermoacoustic instability. However, previous studies have focused on systems with monocritical transitions that switch between only two stable states. In this study, as an example of multistable complex systems, we analyse multiple tipping in a system with a time-dependent parameter and three stable limit cycles under coloured noise excitation. By stochastically averaging the system and performing numerical simulations, multiple tipping-delay phenomena are identified, with the first tipping exhibiting a larger delay. Joint influences of the rate of parameter change and the noise parameters on the tipping behaviours are also discussed. It is revealed that both high rate of change and high noise intensity can turn multiple tipping into single tipping, where the former causes skipping of an intermediate amplitude state and results in direct jumps from a low-amplitude state to a high-amplitude state, while the latter changes the tipping from abrupt to smooth. These findings provide a new perspective for understanding tipping phenomena in multistable systems.