Avalanches and scaling collapse in the large-$N$ Kuramoto model

We study avalanches in the Kuramoto model, defined as excursions of the order parameter due to ephemeral episodes of synchronization. We present scaling collapses of the avalanche sizes, durations, heights, and temporal profiles, extracting scaling exponents, exponent relations, and scaling functions that are shown to be consistent with the scaling behavior of the power spectrum, a quantity independent of our particular definition of an avalanche. A comprehensive scaling picture of the noise in the subcritical finite-$N$ Kuramoto model is developed, linking this undriven system to a larger class of driven avalanching systems.

Figure 1

Definition of avalanche metrics. An avalanche is defined as the section of the order parameter between two consecutive zero crossings. The height $H$ of an avalanche is the maximum height of the order parameter, the size $S$ is the area under the $r\left(t\right)$ curve, and the duration $T$ is the time between the two zero crossings.

Figure 2

(a)–(c) Complementary cumulative distribution functions (CCDF) of the avalanche durations, $T$, sizes, $S$, and heights, $H$, for 13 values of the coupling strength $K$. (d)–(f) Corresponding collapse of the CCDFs by rescaling the $x$ and $y$ axes. The CCDF corresponding to $K=0$ is not included in the collapses because it is not expected to collapse. The number of avalanches observed for each value of $K$ was between 2945 and $10377$. The critical coupling value is $K_C\approx 1.596$. The tails on the left side of the CCDFs correspond to the smallest avalanches, which are distorted due to their durations being comparable to the time step. As such, the long left tails of the CCDFs ought not be expected to collapse.

Figure 3

(a)–(c) Average duration, $T$, size, $S$, and height, $H$, of avalanches for each value of the coupling constant $K$ are shown, where $K_C=1.596$. The constant slope on the log-log plot reveals that there is a power-law relationship between the average values and distance from criticality. The slopes for (a)–(c) are $-\nu z(-\alpha +2)=-0.49,-1/\sigma (-\tau +2)=-1.06$, and $-\rho (-\mu +2)=-0.43$, respectively. (d)–(f) The average square duration, size, and height of the avalanches are shown to be related to $K_C-K$ by a power law. The slopes for (d)–(f) are $-\nu z(-\alpha +3)=-1.2,-1/\sigma (-\tau +3)=-2.3$, and $-\rho (-\mu +3)=-1.0$, respectively. The power-law relationships, as well as the values of the scaling exponents, are given in Table 1.

Figure 4

(a),(b) Average order parameter value as a function of time for avalanches of a certain duration, $T$, and the corresponding collapse. The average was for all avalanches with duration $T$ to $T+T/10$. These curves are collapsed onto a scaling function of $Ax(1-x)$ where $x=t/T$ by rescaling the $y$ axis by a factor of $T^{1-1/\sigma \nu z}$ as shown. (c),(d) The average order parameter value for avalanches of a certain size, $S$, and the corresponding collapse. The average was for all avalanches with size $S$ to $S+S/10$. The curves are collapsed after rescaling the $y$ axis by $S^{1-\sigma \nu z}$ and the $x$ axis by $S^{\sigma \nu z}$. The curve on which the lines collapse in (d) is $Bxexp(-C{x}^2/2)$, where $x=t/S^{\sigma \nu z}$. For these plots $\sigma \nu z=0.54$ as given in Table 1. The avalanches were extracted from simulations with coupling strength $K=1.35$.

Figure 5

Power spectral density of the order parameter for each value of coupling constant $K$. The power-law scaling regime grows as $K$ approaches criticality. The slope of the power law in the scaling regime is $-1/\sigma \nu z=-1.8$ in accord with the values given in Table 1. The roll off at high frequencies appears universal, but is presumably related to the Gaussian distribution of natural oscillator speeds. The inset shows a collapse of the three highest $K$ values using the scaling form from Table 1. The values of the scaling exponents are consistent with those used in the collapse of the avalanches, $1/\sigma \nu z=1.8$ and $\nu z=0.68$. As expected, the collapse works well at low frequencies, where the tested scaling form is predicted to apply.

Figure 6

(a)–(c) Complementary cumulative distribution functions (CCDFs) of the avalanche durations, $T$, sizes, $S$, and heights, $H$, for 12 values of the coupling strength $K$. The number of oscillators simulated here is $N={10}^4$. (d)–(f) Corresponding collapses of the CCDFs obtained by rescaling the $x$ and $y$ axes. The critical coupling value is $K_C\approx 1.596$. The values of the scaling exponents in the collapse are identical to the values used to collapse the CCDFs for the larger system size of $N={10}^6$. The CCDFs closest to the critical coupling strength show finite-size effects, as expected for the smaller system size.

Figure 7

(a)–(c) Average duration, $T$, size, $S$, and height, $H$, of avalanches for each value of the coupling constant from simulations with a system size of $N={10}^4$. The power-law scaling exponents are the same as those found for the larger system size of $N={10}^6$. The slopes for (a)–(c) are $-\nu z(-\alpha +2)=-0.49,-1/\sigma (-\tau +2)=-1.06$, and $-\rho (-\mu +2)=-0.43$ respectively. (d)–(f) The average square duration, $T$, size, $S$, and height, $H$, of avalanches for each value of the coupling constant from simulations with a system size of $N={10}^4$. The slopes for (d)–(f) are $-\nu z(-\alpha +3)=-1.2,-1/\sigma (-\tau +3)=-2.3$, and $-\rho (-\mu +3)=-1.0$, respectively. The power-law relationships, as well as the values of the scaling exponents, are given in Table 1. Finite-size effects are visible close to the critical coupling strength, as expected for the smaller system size.