Nested basins of attraction and noise-induced extreme events in a nonsmooth system

Exploring the mechanisms of extreme events and predicting extreme events are two challenging tasks in different scientific fields. The study of the mechanisms from the perspective of global analysis has received little attention. We make an attempt to establish a link between a type of nested basin structure and occurrence of extreme events. By considering a single-degree-of-freedom spur gear system with nonsmooth factors, it is found that the system exhibits a nested basin of attraction structure for different types of coexisting attractors under certain parameters. The meshing states corresponding to coexisting attractors, including drive-side tooth engaging, tooth disengagement and back-side tooth contacting, are identified. Using the basin boundary saddle order and manifold analysis methods, the complex yet organized structure of the nested basin of attraction is analyzed. We quantify the hierarchy of the nested boundaries through box entropy and evaluate the reliability of the safe operating state using basin stability estimation. The number of nested basins is also analyzed through a series of saddle-node bifurcations. Furthermore, noise-induced multistable transition and extreme events are explored by the probability density function, effective energy landscapes and stability landscape. Different noise intensities induce distinct multistate transitions, which are correlated with the nested basin structure. It is found that the large-amplitude attractors (back-side tooth contacting) located at the outermost layer of the nested basin structure are the primary factors leading to the occurrence of extreme events. Finally, a reservoir computing machine learning is able to successfully predict the transition point to extreme events based on partial information of time series. Overall, the results provide a unifying framework that bridges mechanisms and prediction of extreme events in nonsmooth systems, which helps with the dynamic optimization and design of gear systems from a multidisciplinary perspective.