Power spectrum scaling as a measure of critical slowing down and precursor to tipping points in dynamical systems

Many dynamical systems experience sudden shifts in behaviour known as tipping points or critical transitions, often preceded by the 'critical slowing down' (CSD) phenomenon whereby the recovery times of a system increase as the tipping point is approached. Many attempts have been made to find a tipping point indicator: a proxy for CSD, such that a change in the indicator acts as an early warning signal. Several generic tipping point indicators have been suggested, these include the power spectrum (PS) scaling exponent whose use as an indicator has previously been justified by its relationship to the well-established detrended fluctuation analysis (DFA) exponent. In this paper we justify the use of the PS indicator analytically, by considering a mathematical formulation of the CSD phenomenon. We assess the usefulness of estimating the PS scaling exponent in a tipping point context when the PS does not exhibit power-law scaling, or changes over time. In addition we show that this method is robust against trends and oscillations in the time series, making it a good candidate for studying resilience of systems with periodic oscillations which are observed in ecology and geophysics.