Fig. 3: The numerical verification of the predicted transition.
From: The transition to synchronization of networked systems
a, b The normalized synchronization errors Ecl (see text for definition) as a function of d, for the Lorenz (a) and the Rössler (b) case. Data refers to ensemble averages over 500 different numerical simulations of the network sketched in (a) of Fig. 2. Cluster 1 (yellow line) is formed by nodes {7, 8, 9, 10}, Cluster 2 (orange line) is formed by nodes {4, 5, 6}, and Cluster 3 (red line) is formed by nodes {1, 2, 3}. The black dotted line refers to the synchronization error of the entire network (EN). In both panels it is seen that the expected sequence of events taking place during the transition is verified. Furthermore, the values d1 = 7.322/6 = 1.220 (d1 = 0.179/6 = 0.0298), d2 = 7.322/4 = 1.8305 (d2 = 0.179/4 = 0.04475) and d3 = 7.322 (d3 = 0.179) are marked in the horizontal axis respectively with a yellow, orange and red filled dot in (a) (in b)), indicating how accurate are the predictions and approximations made on the corresponding critical values for the coupling strength. For each interval, the arrow points to the composition of the synchronized cluster that is being observed, once again in perfect harmony with the predictions made. Finally, we have verified that no extra synchronization features emerge during the transition, other than those explicitly foreseen in Fig. 2. b1–b4 Temporal snapshots illustrating the evolution of the y variable of each of the 10 network’s nodes (see color code at the bottom of the four panels) during the transition to synchronization reported in b. At d = 0.01 (b1) the nodes display a fully uncorrelated dynamics. At d = 0.035 b2 the yellow nodes (7,8,9,10) are clustered and display a synchronous motion, whereas all other nodes feature a uncorrelated dynamics. At d = 0.1 (b3) the violet nodes (4,5,6) have joined the clustered evolution, while nodes (1,2,3) remains unsynchronized. Finally, at d = 0.2 (b4), all network’s nodes are synchronized.
