On spatial synchronisation as a manifestation of irregular energy cascades in continuous media under the transition to criticality

Abstract

We investigate the effects of spatial synchronisation occurring in stochastic dynamical systems before extreme events. Unlike the existing studies focused on the mutual coherence of two different time series, we consider multivariate data of the same nature with an arbitrary number of observables, which is typical for problems of continuous media. We show that both Fourier- and wavelet-based coherence analysis methods allow accurate prediction of extreme events independently of the specific nonlinear mechanism(s) driving a complex system to a catastrophe. We also discuss the physical foundations underlying the synchronisation of motion of the system’s constituents on the eve of extreme events by relating the different mechanisms of the transition to criticality through synchronisation with irregularities appearing in the spectral energy cascades.

Appendices

Detrended fluctuation analysis

Under the standard DFA concept, one defines $F(s) \sim s ^ H$, where F(s) is a certain function of data $\{x_k\}_{k = 1}^N$ (the ‘data measure’), while $s\ge 1$ is an arbitrary number called ‘scale’ at which the data are being considered. Specifically, in this research we used the following definitions

$$\begin{aligned} F(s) = \left[ \frac{1}{n}\sum _{k = 1}^{n} \left( R_k^{(s)}\right) ^q\right] ^{1/q},\qquad q\ge 1, \end{aligned}$$

(9)

where

$$\begin{aligned} R_k^{(s)}= & {} \left[ \frac{1}{N/n}\left\| \mathbf {r}_k^{\,(s)}\right\| _p^p\right] ^{1/p}, \qquad p\ge 1, \end{aligned}$$

(10)

$$\begin{aligned} r_{k,m}^{(s)}= & {} \frac{1}{2}\left( \sum _{i = m+\frac{N}{2}}^{m+\frac{N}{2}+s-1}x_i^{(k)} - \sum _{i = m+\frac{N}{2}-s}^{m+\frac{N}{2}-1}x_i^{(k)}\right) , \end{aligned}$$

(11)

$$\begin{aligned} x_i^{(k)}= & {} x_l,\qquad l = (k - 1)\frac{N}{n} + i. \end{aligned}$$

(12)

Here $n = \frac{N}{2s}$, $\mathbf {r}_k^{\,(s)} = \left\{ r_{k,m}^{(s)}\right\}$, $m = 1,\ldots ,\frac{N}{n}+1$. The numbers p and q define specific norms in which the corresponding quantities are to be calculated; these can be chosen arbitrarily, for example, the set $p\rightarrow +\infty$, $q = 1$ originates in the Hurst analysis, whereas the set $p = q = 2$ corresponds to the standard DFA. For details the reader should refer to [33], where (9)-(12) were shown to yield a useful modification of conventional DFA.²

Multivariate Fourier- and wavelet-based coherence analyses

For p-variate stationary data $\{{\mathbf {x}}_k\}_{k = 1}^N = \{(x_k^{(1)}, \ldots , x_k^{(p)})\}_{k = 1}^N$, one defines the partial coherence coefficient between the $(p-1)$-variate $\{(x^{(i)}_k)\}$, $i \ne l$, and the univariate $\{x^{(l)}_k\}$ time series at a frequency f as the maximum eigenvalue $\lambda _l(f)$ of the matrix [34]

$$\begin{aligned} c_l(f) = \frac{{\mathbf {S}}_{{\mathbf {x}}x^{(l)}}^*{\mathbf {S}}^{-1}_{{\mathbf {x}}{\mathbf {x}}}{\mathbf {S}}_{{\mathbf {x}}x^{(l)}}}{S_{x^{(l)}x^{(l)}}}. \end{aligned}$$

(13)

Then, the total coherence is defined as

$$\begin{aligned} C(f) = \left( \prod _{l = 1}^p\lambda _l(f)\right) ^\frac{1}{p}. \end{aligned}$$

(14)

In (13), ${\mathbf {S}}_{{\mathbf {x}}{\mathbf {x}}}(f)$ is the $(p-1)\times (p-1)$ matrix of the power cross- (if $i\ne j$) or proper (if $i=j$) spectra of the scalar time series $\{x^{(i)}_k\}$ and $\{x^{(j)}_k\}$, ${\mathbf {S}}_{{\mathbf {x}}x^{(l)}}(f)$ is the $(p-1)\times 1$ matrix of the power cross-spectra of the scalar time series $\{x^{(i)}_k\}$ and $\{x^{(l)}_k\}$, ${\mathbf {S}}_{{\mathbf {x}}x^{(l)}}^*(f)$ is its conjugate transposed and $S_{x^{(l)}x^{(l)}}(f)$ is the power spectrum (just a real number) of the lth variate at the frequency f.

The spectral matrices in (13) can be computed using diverse spectral analysis methods. In particular, if continuous wavelet transform with a wavelet function $\psi _{a,b}(t):=\frac{1}{\sqrt{a}}\psi \left( \frac{t-b}{a}\right)$ is employed, then the matrix elements $s_{ij} = s_{ij}(f)$’s would additionally depend on the parameter of time shift b, i.e. it would be $s_{ij} = s_{ij}(f,b)$.

From the physical point of view, the elements $s_{ij}$’s of the spectral matrices are the cross- or proper energies of the corresponding harmonics, and hence the eigenvalues $\lambda _l$’s of the resulting matrices $c_l$’s are the energy ratios. Therefore, the total coherence C defined via (14) is the averaged energy ratio (computed as the geometric mean) of all the time series involved, at the frequency f and, if relevant, at the time shift b.

Formulas (13)-(14) are a generalisation of bivariate coherence analysis. Specifically, under $p = 2$ the matrices in the numerator of (13) are simply scalars, the eigenvalues of $c_l$’s trivially coincide with their values, and yet $c_1 = c_2$, so that eventually it holds $C = \sqrt{\lambda _1\lambda _2} = c_1$ (cf. [35]).

Note that in both above-defined DFA and CA methods the data are considered to be finite samples, and thus, applying (9)–(12) and (13)–(14), one obtains the values for H and C(f) (or C(f, b)) within the corresponding time windows of size N. If one repeats the computations in a moving window, then this will yield the time-dependent quantities H(t) and C(f, t) (or C(f, b, t)).

Computer codes implementing the above-described algorithms are available at the GitHub repository [17].

Derivation of power spectrum estimates under VAR(q) coherence model

For a time series whose power spectrum has a linear dependence of the energy S on the frequency $\omega$ (a blue noise), the solution of the Yule–Walker equations yields $a_l\approx \frac{1}{2\sqrt{l}}$. We shall use this approximation for the estimation of the sums in formula (4) at $\omega = 0$ and $\omega = \pi$.

We have to consider four different cases.

Case 1—Mechanism 1, $\omega = 0$. It holds

$$\begin{aligned}&\sum _{l=1}^qa_{21,l}\mathrm {e}^{i\omega l} = \sum _{l=1}^qa_{21,l} \nonumber \\&\quad = \sum _{l=1}^qCa_{22,l} \approx C \int _{1}^{q}\frac{1}{2\sqrt{l}}dl=C(\sqrt{q}-1), \end{aligned}$$

(15)

$$\begin{aligned}&\sum _{l_1=1}^q\sum _{l_2=l_1+1}^qa_{12,l_1}a_{12,l_2}\cos \omega (l_1-l_2)\nonumber \\&\quad \approx C^2\int _{1}^{q}\frac{1}{2\sqrt{l_1}}\nonumber \\&\quad \int _{l_1}^{q}\frac{1}{2\sqrt{l_2}}d l_2 d l_1=\frac{C^2}{4}(2q - 4\sqrt{q} + 2), \end{aligned}$$

(16)

$$\begin{aligned}&\sum _{l=1}^qa_{12,l}^2 \approx C^2\int _{1}^{q}\left( \frac{1}{2\sqrt{l}}\right) ^2dl = \frac{C^2}{4}\ln q. \end{aligned}$$

(17)

Given (15)-(17), for the summands in (4) we find

$$\begin{aligned}&E\frac{E^* + F_1^*\sum _{l=1}^qa_{21,l}}{1-\sum _{l=1}^qa_{22,l}}\sum _{l=1}^qa_{12,l}\nonumber \\&\quad \approx -\frac{|E|^2(1+\sum _{l=1}^qa_{21,l})}{\sum _{l=1}^qa_{22,l}-1}\sum _{l=1}^qa_{21,l}\nonumber \\&\quad \rightarrow -|E|^2C(1 + C(\sqrt{q}-1)), \end{aligned}$$

(18)

$$\begin{aligned}&\quad \frac{E + F_1\sum _{l=1}^qa_{21,l}\mathrm {e}^{-i\omega l}}{1-\sum _{l=1}^qa_{22,l}\mathrm {e}^{-i\omega l}}\frac{E^* + F_1^*\sum _{l=1}^qa_{21,l}\mathrm {e}^{i\omega l}}{1-\sum _{l=1}^qa_{22,l}\mathrm {e}^{i\omega l}}B(\omega )\nonumber \\&\quad \approx \frac{|F|^2(\sum _{l=1}^qa_{21,l})^2+2|E|^2\sum _{l=1}^qa_{21,l}+|E|^2}{(1-\sum _{l=1}^qa_{22,l})^2}\nonumber \\&C^2\left( q-2\sqrt{q}+\nonumber \right. \\&\quad \left. 1+\frac{1}{4}\ln q\right) \rightarrow |F|^2C^4q+|E|^2C^2(2C\sqrt{q}+1). \end{aligned}$$

(19)

Here and below, we used the fact that q can be arbitrarily large, so that we wrote down only the summands at the major exponents of q and ignored those growing slower or even decaying. Substitution of (18)–(19) into (4) subject to $q\rightarrow \infty$ results in the expression shown in Table 2, row 1, column 1.

Case 2—Mechanism 2, $\omega = 0$. It holds

$$\begin{aligned}&\sum _{l=1}^qa_{21,l}\mathrm {e}^{i\omega l} = \sum _{l=1}^qa_{21,l} = \sum _{l=1}^qCa_{22,l}\left( 1-\frac{l}{q}\right) \nonumber \\&\approx C \int _{1}^{q}\frac{1}{2\sqrt{l}}\left( 1-\frac{l}{q}\right) dl=C\left( \frac{2}{3}\sqrt{q}-1+\frac{1}{3q}\right) , \end{aligned}$$

(20)

$$\begin{aligned}&\sum _{l_1=1}^q\sum _{l_2=l_1+1}^qa_{12,l_1}a_{12,l_2}\cos \omega (l_1-l_2)\nonumber \\&\quad \approx C^2\int _{1}^{q}\frac{1}{2\sqrt{l_1}}\left( 1-\frac{l_1}{q}\right) \int _{l_1}^{q}\frac{1}{2\sqrt{l_2}}\left( 1-\frac{l_2}{q}\right) d l_2 d l_1\nonumber \\&\quad =\frac{C^2}{4}\left( \frac{7}{9}q - \frac{8}{3}\sqrt{q} + \frac{8}{9\sqrt{q}} + 1 - \frac{5}{6q} + \frac{2}{9q^2}\right) , \end{aligned}$$

(21)

$$\begin{aligned}&\sum _{l=1}^qa_{12,l}^2 \approx C^2\int _{1}^{q}\left( \frac{1}{2\sqrt{l}}\left( 1-\frac{l}{q}\right) \right) ^2dl\nonumber \\&\quad = \frac{C^2}{4}\left( \ln q-\frac{3}{2}+\frac{2}{q}-\frac{1}{2q^2}\right) . \end{aligned}$$

(22)

From here for the summands in (4), we obtain

$$\begin{aligned}&E\frac{E^* + F_1^*\sum _{l=1}^qa_{21,l}}{1-\sum _{l=1}^qa_{22,l}}\sum _{l=1}^qa_{12,l}\nonumber \\&\quad \approx -\frac{|E|^2(1+\sum _{l=1}^qa_{21,l})}{\sum _{l=1}^qa_{22,l}-1}\sum _{l=1}^qa_{21,l}\nonumber \\&\quad \rightarrow -\frac{2C}{3}|E|^2\left( 1 + C\left( \frac{2}{3}\sqrt{q}-1+\frac{1}{3q}\right) \right) , \end{aligned}$$

(23)

$$\begin{aligned}&\frac{E + F_1\sum _{l=1}^qa_{21,l}\mathrm {e}^{-i\omega l}}{1-\sum _{l=1}^qa_{22,l}\mathrm {e}^{-i\omega l}}\frac{E^* + F_1^*\sum _{l=1}^qa_{21,l}\mathrm {e}^{i\omega l}}{1-\sum _{l=1}^qa_{22,l}\mathrm {e}^{i\omega l}}B(\omega )\nonumber \\&\quad \approx \frac{|F|^2(\sum _{l=1}^qa_{21,l})^2+2|E|^2\sum _{l=1}^qa_{21,l}+|E|^2}{(1-\sum _{l=1}^qa_{22,l})^2}\frac{C^2}{4}\nonumber \\&\quad \left( \frac{14}{9}q-\frac{16}{3}\sqrt{q}+\right. \nonumber \\&\quad \left. \frac{16}{9\sqrt{q}}+2-\frac{5}{3q}+\frac{4}{9q^2}+\ln q-\frac{3}{2}+\frac{2}{q}-\frac{1}{2q^2}\right) \nonumber \\&\quad \rightarrow \frac{14}{81}|F|^2C^4q+|E|^2C^2\left( \frac{14}{27}C\sqrt{q}+\frac{7}{18}\right) . \end{aligned}$$

(24)

Substitution of (23)-(24) into (4) subject to $q\rightarrow \infty$ results in the expression shown in Table 2, row 2, column 1.

Cases 3 and 4—Mechanisms 1 and 2, $\omega = \pi$. These cases are much simpler than the first two, because we can approximately take that the alternating-sign sums containing the terms $\mathrm {e}^{\pm i\omega l}$ and $\cos \omega (l_1-l_2)$ vanish, so that only the summand $\sum _{l=1}^qa_{12,l}^2$ remains. Hence, for the first mechanism we have

$$\begin{aligned} S(\omega )\approx \frac{|E|^2+0+\frac{E+F0}{1}\frac{E^*+F^*0}{1}\left( 0+\frac{C^2}{4}\ln q\right) }{1},\nonumber \\ \end{aligned}$$

(25)

while for the second one it holds

$$\begin{aligned} S(\omega )\approx \frac{|E|^2+0+\frac{E+F0}{1}\frac{E^*+F^*0}{1}\left( 0+\frac{C^2}{4}\left( \ln q-\frac{3}{2}+\frac{2}{q}-\frac{1}{2q^2}\right) \right) }{1}.\nonumber \\ \end{aligned}$$

(26)