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SCIENTIA SINICA Physica, Mechanica & Astronomica, Volume 50 , Issue 1 : 010503(2020) https://doi.org/10.1360/SSPMA-2019-0131

Synchronous patterns in complex networks

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  • ReceivedApr 15, 2019
  • AcceptedMay 24, 2019
  • PublishedSep 18, 2019
PACS numbers

Abstract


Funded by

国家自然科学基金(11875182)


References

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  • Figure 1

    (Color online)For chaotic Logistic map described by equation $x_{n+1}={\boldsymbol~F}(x_n)=4x_{n}(1-x_n)$ and by the coupling function ${\boldsymbol~H}(x)={\boldsymbol~F}(x)$, the variation of the largest Lyapunov exponent, $\Lambda$, calculated from MSF with respect to the generalized coupling strength, $\sigma$. $\Lambda$ is negative in the bounded region $\sigma\in(\sigma_1,\sigma_2)$, with $\sigma_1=0.5$ and $\sigma_2=1.5$. The global synchronization state is stable only when all the transverse modes $\lambda_{2,\cdots,N}$ are locating inside the stable region, as the case shown in the figure.

  • Figure 2

    (Color online)The dynamics of slightly desynchronized network. (a) A snapshot of the distribution of node-synchronization-errors. (b) The time evolution of the node-synchronization-errors. (c) The time evolution of the global network synchronization error, $\delta~x'_{\rm~net}=\sum_i~\delta~x'_i/N$. (d) The distribution of time-averaged node-synchronization-error, $\delta~x'_{iT}=\sum_t~\delta~x'_i/T$. This figure is adopted, with permission, from ref. [40].

  • Figure 3

    (Color online) Identifying synchronous pattern in desynchronized complex networks by the method of eigenvector-based analysis. (a) The same as Figure 2(a), with the $8$ most unstable nodes marked separately. (b) The distribution of the elements $|e_{2,i}|$ associated with the mode of $\lambda_2$. (c) The linear relationship between the time-averaged node-synchronization-error, $\delta~x'_T$, and the eigenvector element, $|e_{2,i}|$. (d) The spatiotemporal evolution of node synchronization errors with the reordered node index $i'$ (by the decreasing order of $|e_{2,i}|$). (e) For coupling strength $\varepsilon=1.05>\varepsilon_2=1.02$, the linear relationship between the time-averaged node-synchronization-error, $\delta~x'_T$, and the eigenvector element, $|e_{N,i}|$. (f) The spatiotemporal evolution of node synchronization errors with the reordered node index $i'$ (by the decreasing order of $|e_{N,i}|$). This figure is adopted, with permission, from ref. [40].

  • Figure 4

    (Color online) The 6-node ring-structure network of rotation symmetry $\mathcal{S}$, on which the partial synchronization state $\mathcal{P}_{\mathcal{S}}=(a,a,b,b,b,a)$ can be observed. In this state, nodes 1, 2 and 6 are completely synchronized, and the synchronous manifold is denoted as $a$; nodes $3$, $4$ and $5$ are synchronized to another manifold, which is denoted as $b$. (b) The structure of a 6-node complex network. The link between nodes $5$ and $6$ has the weight $0.8$, and the weights of other links are set as unity. The network structure possesses three symmetries: the reflection symmetries $\mathcal{S}_1$ and $\mathcal{S}_2$, and the $180^{\circ}$ rotation symmetry $\mathcal{S}_3$.

  • Figure 5

    (Color online) For the same network structure shown in Figure 4(b) and employing chaotic Lorenz oscillator as the local dynamics, the transition of network synchronization with respect to coupling strength. (a) The variation of global-synchronization-error, $\delta$, with respect to coupling strength, $\varepsilon$. Global synchronization is achieved at $\varepsilon_c\approx~13.1$. (b) The variations of the relative synchronization error, $\delta~x_i=\left<~x_i-x_2\right>$, with respect to $\varepsilon$. The network reaches the cluster synchronization state $\mathcal{P}_{\mathcal{S}_1}=(a,a,b,b,c,d)$ when $\varepsilon\in(\varepsilon_1,\varepsilon_c)$, with $\varepsilon_1\approx~8.1$. (c) For $\varepsilon=8.4$, the time evolution of the normalized synchronization error, $\Delta~x_i$, with respect to $\varepsilon$. Cluster synchronization state $\mathcal{P}_{\mathcal{S}_1}$ is emerged at $t\approx~29$. This figure is modifed, with permission, from ref. [63].

  • Figure 6

    (Color online) A schematic plot of the pinning scheme. The lower layer is the original network, which consists of 3 symmetric clusters and supports potentially a 3-cluster synchronous pattern. The pattern, however, is unstable. The upper layer is the control network, which is quotient network associated to the synchronous pattern and its dynamics follows eq. (11). All nodes inside the same cluster are pinned by the same node in the quotient network by the pinning strength $\eta$. The unstable pattern is stabilized when $\eta>\eta&apos;_c$, and is controlled to the dynamics of the quotient network when $\eta>\eta_c$. Both the values of $\eta_c$ and $\eta&apos;_c$ can be analyzed. This figure is adopted, with permission, from ref. [65].

  • Figure 7

    (Color online) Controlling synchronous pattern in the Nepal power-grid network. (a) The network structure. The nodes are partitioned into threenontrivial symmetric clusters ($V_1~=~\{1,2,3,4,5\}$ (red), $V_2~=~\{6,7,8\}$ (blue), and $V_3~=~\{9,10,11,12,13\}$ (green)) and two trivial clusters ($V_4~=~\{14\}$ (pink)and $V_5~=~\{15\}$ (yellow)). (b) The variations of the cluster synchronization errors, $\delta~\hat{x}_l$, with $l~=~1,2,3$, and the network synchronization error,$\delta~\hat{x}_{\rm~net}$, as a function of the coupling strength, $\varepsilon$. Clusters 2, 3, and 1 are synchronized at $\varepsilon_1~\approx~0.4$, $\varepsilon_2~\approx~0.8$, and $\varepsilon_3~\approx~1.1$, respectively. For $\varepsilon~>~\varepsilon_4~\approx~8.9$, the network is globally synchronized. (c) The control network. Controller $l&apos;$ is coupled unidirectionally to all oscillators in cluster $m$ of the original network. (d) For $\varepsilon=~0.3$, the variations of the cluster synchronization errors, $\delta~\hat{x}_l$, as a function of the controlling strength, $\eta$. Synchronization is induced in clusters 2, 3, and 1 at $\eta_1~\approx~13$, $\eta_2~\approx~15$, $\eta_3~\approx~16$, respectively. Dashed lines: the variation of the cluster-controller synchronization errors, $\delta~\hat{x}_l$, with respect to $\eta$. For $\eta>\eta_4\approx~29$, $\delta~\tilde{x}_l=0$, indicating that the synchronous pattern is controlled to the desired synchronous pattern. This figure is adopted, with permission, from ref. [65].

  • Figure 8

    (Color online) Inducing the isolated-desynchronization states in random network of $N$=100 chaotic Lorenzoscillators. The coupling strength is fixed as $\varepsilon$=4.4, with whichthe network is asynchronous in the absence of control. (a) The variation of the time-averagedcluster-synchronization errors, $\left<~\delta~x_{1,2}\right>$, as a function of the pinningstrength $\eta$. $\left<~\delta~x_1\right>$ and $\left<~\delta~x_2\right>$ reach 0 at $\eta_{c1}\approx~0.86$ and $\eta_{c2}\approx~0.91$,respectively. The spatiotemporal evolution of the network underthe pinning strengths (b) $\eta$ = 0.8 and (c) $\eta$ = 0.95. The snapshotsof the network taken at $t$ = 20 for (d) $\eta~=~0.8$ and (e) $\eta~=~0.95$.$\Delta~x_i~=~x_i-\bar{x}$, with $\bar{x}$ the network-averaged state. This figure is adopted, with permission, from ref. [66].

  • Figure 9

    (Color online) Topological control of synchronization pattern by adjust a single link. (a) The structure of the original network, which reaches global synchronization from random initial conditions. (d) The time evolution of the node-synchronization errors, $\Delta~x_i~=~x_i-\bar{x}$. (b) The new network structure when the link between nodes 1 and 3 is removed. (e) The network dynamics is switched from global synchronization to synchronization pattern, $\mathcal{P}_{\mathcal{S}_1}=(a,b,c,b,a)$, associated with reflection symmetry $\mathcal{S}_1$. (c) The network structure obtained by rewiring the link between nodes 1 and 4, which satisfies the reflection symmetry $\mathcal{S}_2$. (f) The network is switched from pattern $\mathcal{P}_{\mathcal{S}_1}$ to pattern $\mathcal{P}_{\mathcal{S}_2}=(a,a,b,c,b)$ after a transient period. This figure is adopted, with permission, from ref. [58].

  • Figure 10

    (Color online)Experimental results on topological control of synchronization pattern in networked neuronal circuits. The local dynamics is represented by chaotic Hindmarsh-Rose circuits, and the original network, with the structure the same as Figure 9(a), is globally synchronized. (a) The new network generated by removing the link between nodes 1 and 3 in the original network. (b) The synchronization relationship between nodes 1 and 5, and nodes 2 and 4. Nodes 1 and 5 are synchronized, and nodes 2 and 4 are synchronized. (c) Nodes 1 and 2 are not synchronized. (d) Nodes 1 and 3 are not synchronized. The network reaches the synchronization pattern $\mathcal{P}_{\mathcal{S}_2}=(a,b,c,b,a)$. This figure is adopted from ref. [58].