Warning signs for non-Markovian bifurcations: colour blindness and scaling laws

Abstract

Warning signs for tipping points (or critical transitions) have been very actively studied. Although the theory has been applied successfully in models and in experiments for many complex systems such as for tipping in climate systems, there are ongoing debates as to when warning signs can be extracted from data. In this work, we shed light on this debate by considering different types of underlying noise. Thereby, we significantly advance the general theory of warning signs for nonlinear stochastic dynamics. A key scenario deals with stochastic systems approaching a bifurcation point dynamically upon slow parameter variation. The stochastic fluctuations are generically able to probe the dynamics near a deterministic attractor to reveal critical slowing down. Using scaling laws near bifurcations, one can then anticipate the distance to a bifurcation. Previous warning signs results assume that the noise is Markovian, most often even white. Here, we study warning signs for non-Markovian systems including coloured noise and α -regular Volterra processes (of which fractional Brownian motion and the Rosenblatt process are special cases). We prove that early warning scaling laws can disappear completely or drastically change their exponent based upon the parameters controlling the noise process. This provides a clear explanation as to why applying standard warning signs results to reduced models of complex systems may not agree with data-driven studies. We demonstrate our results numerically in the context of a box model of the Atlantic Meridional Overturning Circulation.

Theoretical and paleoclimatic evidence for abrupt transitions in the Earth system

“…The condition of additive white noise is less trivial. There have recently been some extensions of the theory to time-correlated noise (Boers 2021, Boettner and Boers 2022, Kuehn et al 2022. If the noise is state-dependent, though, knowledge of the actual dependence is needed to still be able to identify critical slowing down, for example by including the changes in the noise characteristics that are induced by the state dependence.…”

Seeking more robust early warning signals for climate tipping points: the Ratio of Spectra method (ROSA)

“…For example, the Amazon Rainforest, which is of particular interest due to its risk of large-scale vegetation dieback in a potentially hotter and drier climate, experiences forcing which is coherent in space and in time, but is also strongly modulated by the El-Niño Southern Oscillation [29]. White noise forcing is also incompatible with long memory processes [30], such as the effect of sea ice changes on the AMOC [31]. Recently, it was shown [31] that traditional EWS can change their characteristics or disappear entirely when we relax these assumptions on the forcing.…”

“…White noise forcing is also incompatible with long memory processes [30], such as the effect of sea ice changes on the AMOC [31]. Recently, it was shown [31] that traditional EWS can change their characteristics or disappear entirely when we relax these assumptions on the forcing. This motivates creating EWS that do not assume white noise.…”

Systematic analysis and optimization of early warning signals for critical transitions using distribution data

“…In this work, we provide a systematic analysis of the CT framework and its associated EWS, to define their range of applicability and understand why discrepancies have been observed between theoretical predictions and experimental data. 32 , 33 Systems biology is characterized by two main paradigms 34 : one investigating the single details of molecular combinations or regulatory networks, alike to “microstates” in statistical mechanics, 35 and another looking for general analytical models, built upon kinetic theories, to understand complicated biochemical processes in simpler and general terms. 36 The latter allows us to construct classes of systems according to universal routes of dynamical development, regardless of the microscopic details.…”