Detecting, anticipating, and predicting critical transitions in spatially extended systems

Abstract

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Tipping point analysis and its applications

“…The order of the polynomial controls the complexity of the potential. Increasing values of L allow more states to be accommodated; for example, a fourth-order polynomial can capture a system with two states (double-well potential) [Kwasniok, 2018, Kwasniok andLohmann, 2009]. The number of system states is estimated by means of a polynomial fit of the probability density function of the data.…”

“…A simplified version of definition (2.14) would be to only count the relative minima (real wells) in the potential. We opt here for the more comprehensive definition of system states as degenerate potentials have been shown to occur in the context of ice-core records [Kwasniok, 2018, Kwasniok andLohmann, 2009].…”

“…We opt here for the more comprehensive definition of system states as degenerate or nearly degenerate potentials have been shown to occur in the context of ice-core records [Kwasniok, 2018, Kwasniok andLohmann, 2009]. Moreover, even if the system possesses real potential wells, changes 29 in the number of states (bifurcations) tend to be picked up more quickly when using the inflection point definition as this is a more sensitive feature.…”

“…The estimation of the number of system states is augmented by a nonlinear dynamical estimation of the shape of the potential. The numerical method to derive the coefficients of the potential uses the unscented Kalman filter [Julier et al, 2000, Kwasniok, 2018, Kwasniok and Lohmann, 2009, Sitz et al, 2002. The method incorporates measurement noise and model error or inadequacy.…”

Generalized early warning signals in multivariate and gridded data with an application to tropical cyclones

“…where K is a random offset chosen uniformly in the range [0,12], the sine wave models the 12-hourly tidal oscillations. The central pressure is given by p c = 950 + 10η 2 .…”