Choosing observables that capture critical slowing down before tipping points: A Fokker-Planck operator approach

Abstract

Tipping points (TP) are abrupt transitions between metastable states in complex systems, most often described by a bifurcation or crisis of a multistable system induced by a slowly changing control parameter. An avenue for predicting TPs in real-world systems is critical slowing down (CSD), which is a decrease in the relaxation rate after perturbations prior to a TP that can be measured by statistical early warning signals (EWS) in the autocovariance of observational time series. In high-dimensional systems, we cannot expect a priori chosen scalar observables to show significant EWS, and some may even show an opposite signal. Thus, to avoid false negative or positive early warnings, it is desirable to monitor fluctuations only in observables that are designed to capture CSD. Here we propose that a natural observable for this purpose can be obtained by a data-driven approximation of the first non-trivial eigenfunction of the backward Fokker-Planck (or Kolmogorov) operator, using the diffusion map algorithm.

Figures (14)

• Figure 1: Eigenfunctions of the Fokker-Planck operator for the one-dimensional double-well model (\ref{['eq:dw1']}) for a range of parameter values from $\beta=0.05$ to $\beta=0.53$ drawn with increasing contrast. The bifurcation occurs at $\beta \approx 0.544$. The eigenfunctions are estimated by eigendecomposition of the discrete approximation of the FP operator via the scheme of Chang and Cooper CHA70. a- d The first four eigenfunctions of the forward FP operator for $\sigma = 0.25$. e Associated potential $V(x)$. f- h The first three non-trivial eigenfunctions of the backward operator $\phi_{1,2,3}$ (rescaled for each parameter value to have a maximum value of 1). The green dots in g indicate the locations of the saddle point for the respective parameter values. The vertical dashed line is the inflection point of the potential in the shallower well, which is independent of $\beta$.
• Figure 2: a-c Eigenvalues $\{\lambda_1, \lambda_2, \lambda_3, \lambda_4 \}$ of the FP operator for the one-dimensional double-well potential as function of the control parameter $\beta$, for different noise levels $\sigma$. The critical value corresponding to the bifurcation is marked by the vertical dashed line. d Eigenvalues $\{\lambda_1, \lambda_2, ... , \lambda_7 \}$ of the FP operator of the two-dimensional double-well (\ref{['eq:gradient']}) as a function of the control parameter $e$, using the noise level $\sigma = 0.3$. The bifurcation point is marked with the vertical dashed line.
• Figure 3: a-j Forward ( a-e) and backward ( g-j) FP eigenfunctions of the two-dimensional double well model (\ref{['eq:gradient']}) with $\sigma = 0.6$ and control parameter $e=0.5$, computed using the method by Chang and Cooper CHA70. The black contour depicts the level where $\psi_n = \phi_n = 0$. The potential $V(x,y)$ of the system is shown as level sets in ( f). The instanton (computed by the method in KIK20) is drawn in purple, and the basin boundary in green. k-t Same but for the model with $\sigma = 0.3$ and control parameter $e=1.0$, which is closer to the bifurcation at $e\approx 1.73$ compared to the case in panels a-j. Note there is numerical noise due to the very low probabilities that occur at the steepest parts of the potential around the boundaries of the domain. This produces numerical artefacts in the zero contour-line of the eigenfunctions, where erroneously the values in the computed eigenfunctions rapidly alternate in sign.
• Figure 4: Reconstruction of eigenfunctions of the 2-d double-well (\ref{['eq:gradient']}) using the DM algorithm. a-d First four diffusion coordinates $\xi_{1,2,3,4}$ that approximate $\phi_{1,2,3,4}$, obtained at control parameter $e=0.5$ and noise strength $\sigma = 0.6$, and estimated by DM on simulated data sampling the whole phase space. The diffusion coordinates are evaluated at evenly spaced grid points using Eq. \ref{['eq:nystrom']}-\ref{['eq:nystrom2']}. e,f First two non-trivial diffusion coordinates of (\ref{['eq:gradient']}) from simulated data restricted to dynamics that remains in the shallow well, with control parameter $e=0.5$ and noise strength $\sigma = 0.3$.
• Figure 5: a, d Simulated data points of the 2-d double-well (\ref{['eq:gradient']}) in the space spanned by the first three diffusion coordinates, for parameter value $e=0.5$ far from the bifurcation (left) and for $e=1.5$ (right), which is closer to the bifurcation at $e\approx 1.73$. The lower panels b,c,e,f show the same data in two-dimensional projections onto the diffusion coordinates ($\xi_1$,$\xi_2$) and ($\xi_1$,$\xi_3$). We use a lower noise level of $\sigma=0.09$ in order to obtain simulation data restricted to the shallow well when very close to the bifurcation, and thus the relation of $\xi_1$ and $\xi_2$ is different compared to Fig. \ref{['fig:grad2D_eigen_reconstr']}e-f, where $\sigma=0.3$.