Synchronization of fluctuating delay-coupled chaotic networks

We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. Focusing on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone, we compare the synchronization properties of static and fluctuating networks in the regime of large delays. We find that random network switching may enhance the stability of synchronized states. Synchronization appears to be maximally stable when fluctuations are much faster than the time-delay, whereas it disappears for very slow fluctuations. For fluctuation time scales of the order of the time-delay, we report a resynchronizing effect in finite-size networks. Moreover, we observe characteristic oscillations in all regimes, with a periodicity related to the time-delay, as the system approaches or drifts away from the synchronized state.

Figure 1

(a) A $N=30$ network with a clockwise rotating backbone and $p=0.3$, so the number of shortcuts is $N_s=9$. The strength of each link is denoted by its color: black is 1 and gray is $1/2$. Notice that the sum of input links on any node is always 1, as imposed in Eq. (4). (b) Average fraction of the networks having $\text{GCD}=1$ in the $N\times p$ space. For high enough number of units and shortcuts the probability of $\text{GCD}>1$ is negligible.

Figure 2

(a) Spectrum of the adjacency matrix of two SW networks, using $N=500$, and $p=1/5$ (left) and $1/2$ (right). Most of the eigenvalues are contained within a ring whose outer radius scales as $p/4$. (b) spectral gap for different values of $N$ as a function of $p$ of our SW networks, which grows approximately as $p/4$. A more accurate fit, with exponent 1.2, is also shown. Inset: Standard deviation of the eigengap, scaling as the square root of $p$.

Figure 3

Measurement of the synchronization Lyapunov exponent (SLE) along with the spectral gap for ${10}^5$ samples of two SW network ensembles with $N=40$ sites, $\epsilon =0.7$, and $p=0.5$ and $p=0.8$, for two different dynamical systems: Bernoulli and Logistic, with parameters $a=1.1$ and $r=3.577$, respectively. (a) Relation between the SLE and the spectral gap. The black line corresponds to the theoretical expression Eq. (9) for coupled Bernoulli maps with long delays and captures correctly the measured SLE, specially at values close to zero. For the Logistic maps, the correlation is still strong but much more involved. (b) SLE histogram for the same four cases, in log-scale. Notice that for Bernoulli we obtain an approximately Gaussian behavior, with nonzero skewness. For the Logistic case, the SLE are distributed in a much more complicated way. The vertical black bar marks the zero SLE.

Figure 4

Average SLE for our SW networks as a function of both $\epsilon$ and $p$, using 100 samples for each point. (a) Bernoulli map, with $a=1.1$. (b) Logistic map with $r=3.577$. The white line delimits the theoretical synchronization region, Eq. (10), for a static network of Bernoulli maps and eigengap $\mathrm{\Delta }=p/4$.

Figure 5

Synchronization level histories for a fluctuating SW network of Bernoulli maps with different fluctuation times. Parameter values are $a=1.1, N=40, p=0.5, T_d=100$, and $\epsilon =0.7$.

Figure 6

Realization average of the synchronization level, $〈\mathcal{S}\left(t\right)〉$, Eq. (7), over $N_s=1000$ samples, as a function of time for different systems. (a) Bernoulli system, $a=1.1, N=40, p=0.5$, and $\epsilon =0.7$ (on average, nonsynchronizing), for different values of $T_n$. (b) Bernoulli system with $a=1.1, N=40$ sites, $p=0.8$, and $\epsilon =0.47$ (in average, synchronizing), for different values of $T_n$. (c) Logistic system, $r=3.577, N=40, p=0.5$, and $\epsilon =0.4$ (in average, nonsynchronizing). In all cases, $T_d=100$, and we only show time steps that are multiples of 100.

Figure 7

Synchronization regions in the parameter space for fluctuating networks of Bernoulli maps in different regimes ($a=1.1, T_d=100$). The blue x's line denotes the ${\epsilon }^{*}$ values as a function of $p$ for fast fluctuations, $T_n=10$. The white, open-squares line denotes the ${\epsilon }^{*}$ value for very slow fluctuations, $T_n={10}^4$. The dashed line denotes the mean-field approximation given by Eq. (11). For comparison with the static case, we have kept the background colormap corresponding to the static SLE from Fig. 4, and the black continuous line is the theoretical synchronization line from Eq. (10).

Figure 8

Average synchronization for a Bernoulli system with large SLE for different $T_n$ values. The parameters are $a=1.1, \epsilon =0.7, T_d=100, N=40$, and $p=1/2$, but we restrict the sampled networks such that $\mathrm{\Delta }<1/10$. This corresponds to a lower bound for ${\epsilon }^{*}>0.909$. Despite having $\epsilon <{\epsilon }^{*}$, the system synchronizes in the fast fluctuations regime.

Figure 9

Long-term synchronization level Bernoulli map networks of different number of units for a wide range of network switching times. A resynchronizing region appears for intermediate values $T_n\gtrsim T_d$ for smaller networks, but the effect disappears for larger systems. Each point corresponds to the average $\mathcal{S}$ over 100 trials after $2\times {10}^6$ time-steps. In order to verify that we are in a stationary regime, we have checked that the graphic does not qualitatively change when using only ${10}^6$ steps. Parameter values were $a=1.1, T_d=100, \epsilon =0.5$, and $p=1/2$.

Figure 10

(a) Short-time evolution of the average synchronization level, with time in units of $T_d$, for $p=0.5$, for a Bernoulli system with $\epsilon =0.7$ and $T_n=1000$, using different values of $T_d$ and $N$. (b) Long time evolution, with the short period $T_d$ filtered out, and time in units of $T_d^2$. Notice that, in all cases, the oscillations have the same frequency and phase. (c) Average synchronization level for two fixed networks with $N=20, p=0.7$, and $\epsilon =0.83$. The ascending greenish sequence corresponds to a synchronizing instance, while the descending reddish sequence refers to a nonsynchronizing one. In both cases, we have averaged over random initial perturbations with amplitude $A\times {10}^{-10}$. Notice that, in all cases, we obtain oscillations with the same frequency and phase but different amplitudes. The Bernoulli slope is $a=1.1$ in all cases.

Figure 11

Delay echoes of a point-like perturbation applied at $t=0$ along the transverse direction $v_2$, in a network of delay-coupled Bernoulli maps. We show the synchronization level $-ln|v_2\left(t\right)|$ immediately after applying the perturbation, and the 10th, 20th, 30th, 40th, and 50th delay echo. We normalized with respect to the initial amplitude. Parameters are $T_d=100, a=3/2, \epsilon =2/3, {\gamma }_2=2/5$.