Extreme events in dynamical systems and random walkers: A review.

Extreme events gain tremendous attention due to their utmost importance in a variety of diverse contexts ranging from climate science to neuroscience. Excursions of a relevant observable from its long-term average to extraordinary values have the capability of bringing adverse consequences. We provide here a comprehensive review to incorporate the recent efforts in understanding such extremely large-amplitude events from the perspective of dynamical systems and random walkers. We emphasize, in detail, the mechanisms responsible for the emergence of such events in the complex systems. We discuss the prediction of extreme events from two different contexts using (i) dynamical instabilities and (ii) machine learning algorithms. Tracking of instabilities in the phase space is not always feasible and precise knowledge of the dynamics of extreme events does not necessarily help in forecasting extreme events. Moreover, in most studies on high-dimensional systems, only a few degrees of freedom participate in extreme events' formation. Thus, the notable inclusion of prediction through machine learning is of enormous significance, particularly for those cases where the governing equations of the model are explicitly unavailable. Besides, random walk on the complex networks is capable of representing several transport processes, and exceedances of the flux of walkers above a prescribed threshold may describe extreme events. We unveil the theoretical studies on random walkers with their enormous potential for applications in reducing extreme events and also discuss possible controlling strategies. This review presents an overview of the current trend of research on extreme events in dynamical systems and networks, including random walkers, and discusses future possibilities. We conclude this review with the extended outlook and compelling perspective for further investigation.