Synchronous patterns in complex systems

When a complex network is slightly desynchronized, a few of the network nodes will be escaping from the uniform synchronization background frequently with a random fashion, leading to the intermittent network synchronization. Here, based on the eigenvectors of the network coupling matrix, we propose a new method which is able to figure out the unstable nodes in the general case of desynchronized complex networks. Moreover, with this method, we are also able to regulate the seemingly random network dynamics into stable and visible synchronous patterns. The efficiency of this method is verified by a variety of network models, including varying the network structures, the node local dynamics, and the desynchronization types. Our studies show that, even for the complex network systems, synchronous patterns can still be identified and characterized.

Figure 1

The intermittent synchronization of the networked chaotic logistic maps. The network is generated according to the BA growth algorithm, which contains $N=800$ nodes and the average degree is 14. The uniform coupling strength is $\varepsilon =0.935$, with which only the 2nd mode is slightly destabilized, ${\Lambda }_1\approx 6\times {10}^{-3}>0$. (a) A snapshot of the synchronization errors, $\delta x_i^{\prime }$, where the size of the nodes is set to be proportional to the amplitude of the synchronization error, while the colors are used for eye guiding. (b) After a transient period of $t=1\times {10}^3$, the time evolution of the synchronization errors, $\delta x^{\prime }$, for all the network nodes. (c) The variation of the network-averaged synchronization error, $\delta x_{\mathrm{net}}^{\prime }$, as a function of time, which experiences the process of on-off intermittency. (d) The distribution of the time-averaged synchronization error, $\delta x_{iT}^{\prime }$, where each data is averaged over a period of $T=1\times {10}^3$. Our tasks in the present work are just to figure the unstable nodes in the complex network of (a), and identify the synchronous patterns in the seemingly random dynamics of (b).

Figure 2

For the same network dynamics as plotted in Fig. 1 ($\varepsilon =0.935$ and only the 2nd mode is slightly destabilized), (a) the replotted snapshot of the node synchronization errors, $\delta x_i^{\prime }$, (b) the distribution of the eigenvector element for the destabilized mode, ${\mathbf{e}}_{2i}$, (c) the variation of the eigenvector element, $|e_{2,i}|$, as a function of the time-averaged synchronization error, $\delta x_{iT}^{\prime }$, and (d) after reordering the network nodes, the time evolution of the synchronization errors. The 8 nodes which have the largest synchronization errors in (a) correspond to exactly the 8 nodes which have the largest eigenvector element in (b). By $\varepsilon =1.05$ (only the $N$th mode is slightly destabilized), (e) the variation of the eigenvector element, $|e_{N,i}|$, as a function of $\delta x_{iT}^{\prime }$, and (f) the time evolution of the synchronization errors with the network nodes be reordered according to the amplitude of the eigenvector elements. For both (c) and (e), we have numerically the relationship $|e_i|\propto \delta x_T^{\prime }$, which agrees with the theoretical predication of Eq. (6).

Figure 3

The eigenvector of the 2nd eigenmode and the relationship between the eigenvector element, $|e_{2,i}|$ and the node synchronization error, $\delta x_{iT}^{\prime }$, for a 2-cluster network [(a),(b)], a 3-cluster network [(c),(d)], and the cat-brain network [(e),(f)]. (g) For the slightly desynchronized cat-brain network, the evolution of the system dynamics with the original node index. (h) The evolution of the system dynamics with the node index be reordered according to the improved method.

Figure 4

For the network of non-identical chaotic Rössler oscillators, the time evolution of the node synchronization errors with (a) the original node index and (b) the reordered node index. In (a), the 5 (interrupted) horizontal lines occur at the nodes of the differentiated parameter, ${\omega }^{\prime }$. In plotting (b), the 5 nodes of the parameter ${\omega }^{\prime }$ are given the smallest indices, 1 to 5, while the resting nodes are reordered according to the eigenvector element, $|e_{2,i}|$, as did in Figs. 2 and 3.