Extreme events in dynamical systems and random walkers: A review

Abstract

Extreme events gain the attention of researchers due to their utmost importance in various contexts ranging from climate to brain. An observable that deviates significantly from its long-time average will have adverse consequences for the system. This brings such recurrent events to the limelight of attention in interdisciplinary research. There is a need for research efforts in many systems in the real world to find solutions that can predict and mitigate the unfavorable effects of these recurring events. A comprehensive review of recent progress is provided to capture recent improvements in analyzing such very high-amplitude events from the point of view of dynamical systems and random walkers. We emphasize, in detail, the mechanisms responsible for the emergence of such events in complex systems. Several mechanisms that contribute to the occurrence of extreme events have been elaborated that investigate the sources of instabilities leading to them. In addition, we discuss the prediction of extreme events from two different contexts, using dynamical instabilities and data-based machine learning algorithms. Tracking of instabilities in the phase space is not always feasible and a precise knowledge of the dynamics of extreme events does not necessarily help in forecasting extreme events. Moreover, in most of the studies on high-dimensional systems, only a few degrees of freedom participate in extreme events’ formation. Thus, a 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 walks on complex networks can represent several transport processes, and exceedances of the flux of walkers above a prescribed threshold may describe extreme events. We unveil theoretical studies on random walkers with their enormous potential for applications in reducing extreme events. We cover the possible controlling strategies, which may be helpful to mitigate extreme events in physical situations like traffic jams, heavy load of web requests, competition for shared resources, floods in the network of rivers, and many more. 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 an extended outlook and compelling perspective, along with the non-trivial challenges for further investigation.