Early-warning indicators for tipping points

Abstract

The term ‘tipping event’ is used to describe a certain class of phenomena as observed in many different fields of science. It refers to an event where a gradual change of external forcing causes a sudden, large, often unwanted, transition to the state of the system. Some examples of known tipping events in science include: Arctic sea ice melting (climate), epileptic seizures (biology), collapse of ecosystems and populations (ecology) and market crashes (finance). Three mathematical mechanisms for tipping events have been proposed in the literature: bifurcation-, noise- or rate-induced tipping. Recent research has focused on developing early-warning indicators to potentially offer forewarning, which can extract from output time series whether the external forcing approaches a critical level at which tipping occurs. Two commonly used early-warning indicators are an increase of autocorrelation and variance in the time series data for the system’s output. The theory behind the presence of these indicators is the loss of stability of the system’s current state known as ‘critical slowing down’ for the approach of a bifurcation-induced tipping. Rate-induced tipping occurs when the external forcing reaches a critical rate instead of level. For rate-induced tipping there is no loss of stability of the system’s current state and therefore it is not clear if the early-warning indicators should exist. In this thesis we investigate the presence of early-warning indicators for models that show rate-induced tipping with additive noise. We also explore a technique for determining the most likely time of tipping using optimal paths for escape. Research has mainly focussed on testing the early-warning indicators for examples of known tipping events in the past. The ultimate aim of early-warning indicators would be to have the ability to predict future tipping events. Using the early-warning indicators in isolation is susceptible to incurring false alarms and missed alarms. We present a method for approximating the probability of experiencing rate-induced tipping with noise for slow to moderate drift speeds.