Anticipating Bifurcations for Identifying Dynamic Characteristics of Complex Systems

Abstract

Complex systems are at risk of critical transitions when the system shifts abruptly from one state to another when a threshold is crossed. Recent studies have revealed that a variety of systems, ranging from systems examined by engineering, physics, and biology, to others related to climate sciences, medicine, social sciences, and ecology are susceptible to transitions leading to drastic re-organization or collapse. Such an unexpected transition is usually undesirable, because it is often difficult to restore a system to its pre-transition state once the transition occurs. It is exceedingly difficult to know if a system comes close to critical transitions because typically no easily noticeable changes can be observed unless the transition happens. Furthermore, accurate models are often not available, and predictions based on models of limited accuracy face difficulties. Hence, we are still ill-equipped to predict critical transitions, and there is an acute need for reliable methods to predict such catastrophic events. In this research, a data-driven, model-free approach is introduced to forecast critical points and post-critical dynamics of complex dynamical systems using measurements of the system response collected only in the pre-transition regime. Based on observations of the system response to perturbations only in the pre-transition regime, the method forecasts the bifurcation diagram and discovers the system’s stability after the transition. The forecasting approach is based on the phenomenon of critical slowing down, referring to the slowing down of a system's dynamics when approaching a tipping point. The rate of the system’s recovery from perturbations decreases when the system approaches the transition. Thus, the rate of recovery from perturbations can be used as an indicator, and is correlated to the distance to the transition. The method is employed to forecast critical transitions in several classes of complex systems including flutter instabilities in fluid-structural systems, collapse of natural populations in ecological systems, and the onset of traffic congestions in vehicular traffic flow systems. The theoretical and experimental results of this study address important challenges in forecasting safety and stability of complex systems. The capabilities of the methods proposed make them unique tools for analysis of complex systems in both computational and experimental studies.