Final state sensitivity and noise-induced extreme events in a nonsmooth dynamical system

Final state sensitivity and extreme events are two phenomena of fundamental importance in nonlinear science, often manifesting unexpectedly across a wide range of scientific disciplines. However, the relationship between these phenomena has received little attention, especially in nonsmooth gear systems, where it has not yet been reported. In this work, we focus on a quantitative characterization of final state sensitivity and on elucidating the mechanism of noise-induced extreme events in a nonsmooth gear system. Three safety levels, corresponding to drive-side tooth engagement, tooth disengagement, and back-side tooth contact, are classified as safe attractors, quasi-safe attractors, and unsafe attractors. To quantify the sensitivity of these three meshing states, we employ different metrics such as basin entropy, boundary basin entropy, and the Wada index. Previously, we employed the grid method and the Nusse–Yorke method to verify the existence of a common basin boundary, and initial conditions located on this boundary occupy most of the phase space. Under the influence of very small noise, trajectories starting from the same initial condition may converge to different meshing states, reflecting strong uncertainty. For weak noise, both quantitative and qualitative aspects of final state sensitivity are analyzed using probability basins and the noise-sensitivity exponent. We calculate the minimum noise amplitude required for trajectories to escape from each attractor into the common basin boundary and identify a range of intermediate noise intensities that tend to drive the system toward safe attractors. As the noise intensity exceeds a critical threshold, extreme events emerge. Their mechanisms are explored through noise-induced intermittency, revealing how noise can trigger rare but significant deviations in system dynamics. Finally, the results provide insight into the different meshing state transitions observed in gear systems, and the research methods proposed in this study can be extended to a broader class of nonsmooth gear systems.