Detecting critical transitions in the case of moderate or strong noise by binomial moments

Detecting critical transitions in the case of moderate or strong noise (collectively referred to as big noise) is challenging, since such noise can make a critical transition point far from the bifurcation point, leading to the failure of traditional small-noise methods. To handle this tough issue, we first transform a generic noisy system into a linear set of binomial moment equations (BMEs). Then, we can solve a closed set of BMEs obtained by truncation and use the resulting binomial moments to reconstruct a joint probability distribution of the state variables of the original system. Third, we derive a leading indicator from the closed set of BMEs. Importantly, the reconstructed distribution determines the way of critical transition (i.e., critical transition is distribution transition rather than state transition in the strong-noise case) as the system comes close to the critical transition point, whereas the derived indicator anticipates when the distribution transition occurs. Our theory has broad applications, and artificial and data examples exhibit its power.

Figure 1

Convergence of moments and approximation of probability distribution [17]. (a) BMs converge to zero, as the moment orders go to infinity. (b) Error analysis and probability distributions reconstructed by BMs, where “error” represents the Kullback-Leibler divergence between two probabilities: the one calculated according to Eq. (11) with the BMs obtained by solving Eq. (9) and the other obtained by directly solving Eq. (3); and “Prob.” in the inset represents probability density function.

Figure 2

Schematic illustration for the characteristics of critical transitions in the system described by Eq. (13). This system is assumed to undergo a fold bifurcation as the parameter $c$ increases in a manner described in a multi-scale simulation approach [16]. (a) State transition in the case of small noise, where the trajectory tends to switch once from the “living” state to the “dying state” at a certain value of parameter $c_1$. This value is close to the bifurcation point $c_0$. (b) Distribution transition in the case of big noise, where the trajectory tends to have large fluctuations and the region can be characterized by its fluctuation property. (c) Characteristic of distribution in three regions: resilient region where the distribution has a peak away from zero, flickering region where the distribution is apparently bimodal, and irresilient region where the distribution has a peak near zero. In (a) and (b), solid blue and dashed red curves are the bifurcation curves of the deterministic systems.

Figure 3

Relationship between dimension raise and noise reduction: the original dynamics of the system with big noise (a) can be embedded into the state dynamics of a higher-dimensional binomial moment system with small noise (b). (c) State transitions in binomial moment system indicate distribution transitions in the original system. This illustration is based on Eq. (13) and the parameter values are set as $h=3$, $v=0.5$, $r=4$, $c=3$, $D=1$.

Figure 4

Characteristics of critical transition before (i.e., the left-hand side of dashed line) and after (i.e., the right-hand side of dashed line) bifurcation: (a) and (c) small noise case, where (a) shows the influence of noise on state whereas (c) shows the dependence of characteristic return time on bifurcation parameter; (b) and (d) big noise case, where (b) shows the influence of noise on state whereas (d) shows the dependence of characteristic return time on bifurcation parameter. Dashed lines represent bifurcation points. The lines corresponding to “small” are obtained by calculating $T_{\mathrm{small}\mathrm{noise}}\left(h\right)$, whereas the lines corresponding to “big” are obtained by calculating $T_{\mathrm{big}\mathrm{noise}}\left(h\right)$. Parameter $h$ is linearly increased from −0.5 to 2.5 as time goes from 0 to 100. Here, $D=0.1$ represents the small noise whereas $D=2.5$ represents the strong noise.

Figure 5

The influence of the truncation order on the characteristic return time of the BME system for three different noise intensities: (a), (c) NBMEs are truncated up to order 4; (b), (d) NBMEs are truncated up to order 5. Panels (a) and (b) show that with the increase of parameter $h$, the system gradually comes across different regions. Panels (c) and (d) show the dependence of $T_{\mathrm{big}\mathrm{noise}}$ on both parameter $h$ and noise intensity $D$.

Figure 6

Comparison of the spectrum between the binomial moment equation and the Fokker-Planck operator. As the increase of parameter $h$, the system comes closer to the distribution transition region and relaxes slower to the ergodic state, which is reflected by the smaller module of ${\mu }_1$ given by the Fokker-Planck operator and ${\lambda }_1$ given by the binomial moment equation (because the spectrum is negative). The spectrum of the Fokker-Planck operator can be computed by Galerkin method [32].

Figure 7

The ability of binomial moments (BMs) to predict critical transition in an eutrophic lake: (a) One-dimensional gradients of sediment diatom composition and diversity based on detrended correspondence analysis (DCA axis 1 scores) (blue) and calculated Hill's diversity index N2 (HDI) (blue), where the data is interpolated. It is believed that a critical transition occurs at around 2001 (marked by dashed line). (b) The first two order BMs of data (a), obtained by estimation using a 59-year (half time series) sliding window of the period 1883–2001. (c) The first three order BMs of data (a), obtained by the same method as (b).