Internal noise interference to warnings of tipping points in generic multi-dimensional dynamical systems

Abstract

A deterministic dynamical system that slowly passes through a generic fold-type (saddle-node) bifurcation can be reduced to one-dimensional dynamics close to the bifurcation because of the centre manifold theorem. It is often tacitly assumed that the same is true in the presence of stochasticity or noise so that, for example, critical slowing down (CSD) indicators can be applied as if the system were one-dimensional. In this work, we show that this is only true when given suitable system observables; specifically, we demonstrate that noise in other dimensions may interfere with indicators of CSD, also referred to as early warning signals (EWS). We point out a generic mechanism by which both variance and lag-1 autocorrelation [AC(1)], as well as other EWS, can fail to signal an approaching bifurcation. This can, in principle, occur whenever one noise source drives multiple system components simultaneously. Even under the favourable assumptions of uncoupled deterministic dynamics and stationary noise, some system observables can then exhibit false negative or false positive CSD indications. We isolate this phenomenon in an example that represents a generic two-dimensional fold-type bifurcation setting.