The Effect of Additive and Multiplicative Cyclic Perturbations on Noise-Induced Tipping Dynamics

The dynamics of systems near tipping points attract considerable attention in the context of climate change, ecological regime shifts, disease spreading, and other complex systems undergoing transitions. In particular, the duration and cause of transitions between states remain subjects of ongoing debate. We address these questions by applying the large-fluctuation framework to analyse noise-induced transitions in a widely studied tipping model describing dynamics near a fold bifurcation. As complex systems are typically not in equilibrium, we include cyclic perturbations representing, for example, diurnal variations, seasonal cycles, solar activity oscillations, and Milankovitch cycles in the climate system. We investigate how the frequency and type of cyclic perturbation influence noise-induced transitions between states by examining the fluctuational force. Two types of periodic perturbations, additive and multiplicative, representing B- and R-tipping, are considered. We show, first, that depending on the type of cyclic perturbation, the fluctuations need to be synchronised with different perturbation phases to induce the transition. Secondly, we demonstrate that the transition duration depends on the perturbation frequency: when the periodic perturbation is slower than the system's relaxation rate, the transition occurs within a single oscillatory cycle, whereas high-frequency perturbations can significantly prolong the transition time.