Efficacy of dynamic eigenvalue in anticipating and distinguishing tipping points

The presence of tipping points in several natural systems necessitates having improved early warning indicators to provide accurate signals of an impending transition to a contrasting state while also detecting the type of transition. Various early warning signals (EWSs) have been devised to forecast the occurrence of tipping points, also called critical transitions. Dynamic eigenvalue (DEV) is a recently proposed EWS that can not only predict the occurrence of a transition but also certain types of accompanying bifurcations. Here, we study the effectiveness and limitations of DEV as an EWS for diverse kinds of critical phenomena. We demonstrate that DEV is a powerful EWS that shows promising results in anticipating catastrophic (first-order or discontinuous) and non-catastrophic (second-order or continuous) transitions in discrete and continuous dynamical systems. However, it falls short in the case of piecewise smooth systems. Further, the ability of DEV to forecast the type of transition is limited, as it cannot differentiate saddle-node bifurcation from transcritical and pitchfork bifurcations. Despite these limitations, we show that DEV can work as a robust indicator for varying rates at which the transition is approached and with systems involving colored noise.