Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators

In a recent study about chaos synchronization in complex networks [Nat. Commun. 5, 4079 (2014)], it is shown that a stable synchronous cluster may coexist with vast asynchronous nodes, resembling the phenomenon of a chimera state observed in a regular network of coupled periodic oscillators. Although of practical significance, this new type of state, namely, the isolated-desynchronization state, is hardly observed in practice due to its strict requirements on the network topology. Here, by the strategy of pinning coupling, we propose an effective method for inducing isolated-desynchronization states in symmetric networks of coupled chaotic oscillators. Theoretical analysis based on eigenvalue analysis shows that, by pinning a group of symmetric nodes in the network, there exists a critical pinning strength beyond which the group of pinned nodes can completely be synchronized while the unpinned nodes remain asynchronous. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks with the numerical results in good agreement with the theoretical predictions.

Figure 1

Inducing isolated desynchronization from a six-node asynchronous network of coupled chaotic Lorenz oscillators. (a) The network structure. Nodes are grouped into two clusters according to their symmetries: $V_1=\{1–4\}$ (rotation symmetry) and $V_2=\{5,6\}$ (reflection symmetry). (b) By the coupling strength $\varepsilon =0.7$ the time evolution of the cluster-synchronization errors $\delta x_{1,2}$ in the absence of the pinning control. Both clusters are asynchronous. (c) With the pinning strength $\eta =8.0$ the time evolutions of $\delta x_{1,2}$. Cluster 2 is synchronized at about $t=15$, whereas cluster 1 remains asynchronous throughout the process. (d) The variation of the time-averaged cluster-synchronization errors $\langle \delta x_{1,2}\rangle$ as a function of $\eta .\langle \delta x_2\rangle \approx 0$ at ${\eta }_c\approx 7.8.\langle \delta x_c\rangle$ is the time-averaged synchronization error between oscillator 5 and the controller; ${\langle \delta x\rangle }_{min}$ is the smallest synchronization error between oscillator 5 and oscillators in cluster 1.

Figure 2

Inducing isolated desynchronization in the network of the Nepal power grid. The nodal dynamics and coupling function are the same as in Fig. 1. The coupling strength is fixed as $\varepsilon =0.32$ with which no synchronization is established between any pair of the oscillators. Pinning control is added on oscillators in cluster 1. (a) The variation of the time-averaged synchronization error of cluster 1 $\langle \delta x_1\rangle$ as a function of the pinning strength $\eta$. Isolated desynchronization is induced when $\eta >{\eta }_c\approx 16$. The spatiotemporal evolution of the oscillators under the pinning strengths (b) $\eta =14$ and (c) $\eta =18$.

Figure 3

Inducing the isolated-desynchronization state of two synchronous clusters in a random network of $N=100$ chaotic Lorenz oscillators. The coupling strength is fixed as $\varepsilon =4.4$ with which the network is asynchronous. (a) The variation of the time-averaged cluster-synchronization errors $\langle \delta x_{1,2}\rangle$ as a function of the pinning strength $\eta .\langle \delta x_1\rangle$ and $\langle \delta x_2\rangle$ reach 0 at ${\eta }_{c1}\approx 0.86$ and ${\eta }_{c2}\approx 0.91$, respectively. The spatiotemporal evolution of the network under the pinning strengths (b) $\eta =0.8$ and (c) $\eta =0.95$. The snapshots of the network taken at $t=20$ for (d)$\eta =0.8$ and (e) $\eta =0.95$. $\mathrm{\Delta }{x}_i=x_i-\overline{x}$ with $\overline{x}={\sum }_i{x}_i/N$ as the network-averaged state.