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Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method

  • * Corresponding author: Wenxue Li

    * Corresponding author: Wenxue Li
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  • We consider the outer synchronization between drive-response systems on networks with time-varying delays, where we focus on the case when the underlying networks are not strongly connected. A hierarchical method is proposed to characterize large-scale networks without strong connectedness. The hierarchical algorithm can be implemented by some programs to overcome the difficulty resulting from the scale of networks. This method allows us to obtain two different kinds of sufficient outer synchronization criteria without the assumption of being strongly connected, by combining the theory of asymptotically autonomous systems with Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory. The theory improves some existing results obtained by graph theory. As illustrations, the theoretic results are applied to delayed coupled oscillators and a numerical example is also given.

    Mathematics Subject Classification: Primary: 34A34; Secondary: 68W01, 34C15.


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  • Figure 1.  Example of two networks with identical coupling structure. The solid lines represent the deterministic coupling among the nodes within a network, while dashed ones represent the coupling between the drive network and response network.

    Figure 2.  A general digraph $\mathcal{G}$ with 67 vertices and the corresponding condensed digraph $\mathcal{H}$

    Figure 3.  Digraph $(\mathcal{G},B)$ with 20 vertices and a not strongly connected network.

    Figure 4.  The pathes of the derive system (7) with initial value $(x_{1+5i} = 50,\tilde{x}_{1+5i} = -50,x_{2+5i} = 60,$$\tilde{x}_{2+5i} = -60,x_{3+5i} = 0,\tilde{x}_{3+5i} = 0,x_{4+5i} = 60,$$\tilde{x}_{4+5i} = -60,x_{5+5i} = 70,\tilde{x}_{5+5i} = -70,i = 0,1,2,3)$.

    Figure 5.  The pathes of response system (8) with initial value $(y_{1+5i} = 0,\tilde{y}_{1+5i} = -170,y_{2+5i} = -70,$$\tilde{y}_{2+5i} = 30,y_{3+5i} = 0,\tilde{y}_{3+5i} = 0,y_{4+5i} = -30,$$\tilde{y}_{4+5i} = 30,y_{5+5i} = 40,\tilde{y}_{5+5i} = 20,i = 0,1,2,3)$.

    Figure 6.  The pathes of the synchronization error system with initial value $(e_{1+5i} = -50,\tilde{e}_{1+5i} = -120,e_{2+5i} = -130,$$\tilde{e}_{2+5i} = 90,e_{3+5i} = 0,\tilde{e}_{3+5i} = 0,e_{4+5i} = -100,$$e_{5+5i} = -40,\tilde{e}_{5+5i} = 110,i = 0,1,2,3)$.

    Figure 7.  Digraph $(\mathcal{G},A)$ and its strongly connected components $\mathfrak{H}_{i}$ are shown in (a). The corresponding condensed digraph $\mathcal{H}$ is shown in (b).

    $\mathrm{Forms}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$
    $\tau_{k}(t)$ $0.5(\sin(t)+1)$ $0.5(\cos(t)+1)$ $0.4(\sin(t)+1)$ $0.4(\cos(t)+1)$
    $ N_{h}(x_{h})$ $\sin(x_{h})$ $\cos(x_{h})$ $0.5\sin(2x_{h})$ $0.5\cos(2x_{h})$
     | Show Table
    DownLoad: CSV
    $\mathrm{Parameters}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$
    $\epsilon_{k}$ $1.8$ $1.9$ $2.0$ $2.1$
    $ \varepsilon_{k}$ $0.05$ $0.1$ $0.15$ $0.2$
     | Show Table
    DownLoad: CSV
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