Developing an Early Warning Indicator for strongly forced systems far from equilibrium: increased variance of the quasi-equilibrium

Abstract

Early Warning Indicators (EWIs) have been developed in an attempt to forewarn of approaching tipping points. In the case of bifurcation-induced tipping (Ashwin et al., 2012), the most commonly used EWIs are related to the phenomenon of 'Critical Slowing Down (CSD)', which involves an increase in the variance and autocorrelation of a state variable prior to the bifurcation (Scheffer et al., 2009). CSD can provide useful early warning in systems which remain close to a quasi-equilibrium state as they approach the bifurcation (Dakos et al., 2008). Under these circumstances, fluctuations reveal the declining stability of the steady state. However, CSD is far less reliable in a strongly forced system where the control variable is changing quickly compared to the characteristic timescale of the system. In such strongly forced systems, it is possible to temporarily overshoot the bifurcation, without causing tipping (Ritchie et al., 2021). However, currently there is no robust way of determining when a strongly forced system has passed the bifurcation. Here we develop a more reliable EWI for strongly forced systems, by looking instead at the fluctuations about a reconstructed quasi-equilibrium state. The variance of the reconstructed quasi-equilibrium state blows-up as the bifurcation point is approached, such that the inverse of the variance goes to zero at the bifurcation. We show that our new EWI, which we call 'Increased Variance of Quasi-equilibrium (IVQ)', is able to forewarn of a critical point being crossed in both weakly and strongly forced systems.