Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation.

Figure 1

Synchronization of coupled logistic maps for different values of coupling strength $\epsilon$ and connection delays $\tau$, for the cases of (a) scale-free, (b) random, (c) small-world, and (d) nearest-neighbor coupling. The gray scale encodes the degree of synchronization, with black regions corresponding to complete synchronization.

Figure 2

The bifurcation diagram of the synchronized solution of coupled logistic maps for several values of $\tau$. At each value of $\rho$, 200 iterates of (3) are plotted after initial transients. The value of $\epsilon$ is 0.8.

Figure 3

The Lyapunov exponents of the synchronized solution for $\tau =1$ and $\epsilon =0.8$.

Figure 4

The synchronized solution at different coupling strengths for $\tau =1$ and $\rho =4$. At each value of $\epsilon$, 300 iterates of (3) are plotted after initial transients.

Figure 5

Intermittent behavior of synchronized logistic maps for $\rho =4$, $\epsilon =0.95$, and $\tau =1$.