Early warnings of critical transitions through vector autoregression: lessons from multiscale systems

Abstract

In a nonautonomous nonlinear dynamical system, generic critical transitions (tipping points) are not limited to slow passage through fold bifurcations. They can also correspond to slow passage through other generic bifurcations, such as subcritical Hopf or even (in the multiscale setting) singular Hopf bifurcation. In systems such as these, generic early warning signals associated with critical slowing down, such as observing an increase in variance and lag-1 autocorrelation, can be obscured by oscillatory behaviour. Here, we present a method using vector autoregression (VAR) that aims to identify multiple eigenvalues of the system from multiple time series. With this, we are not only able to deduce the system's stability but also identify the type of bifurcation we are approaching and, therefore, the underlying dynamics, which gives us insight into the system's future state post-tipping. In practice, this can be beneficial for estimation and mitigation of risk. We look at three bifurcations in particular: the fold, the subcritical Hopf bifurcation, and the singular Hopf bifurcation in the presence of an extra time-scale separation. We hope to show that VAR is potentially a powerful tool in an EWS toolkit and can be more enlightening than simply looking at AR(1) of individual time series, especially if there is more than one timescale present in the system.