# American Institute of Mathematical Sciences

March  2018, 23(2): 837-859. doi: 10.3934/dcdsb.2018045

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

 1 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China 2 Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

* Corresponding author: Wenxue Li

Received  December 2016 Revised  September 2017 Published  December 2017

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.

Citation: Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045
##### References:
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Bi, Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 284-291.  doi: 10.1016/j.cnsns.2010.11.029.  Google Scholar [36] Q. Zhu and J. Cao, pth moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching, Nonlinear Dyn., 67 (2012), 829-845.  doi: 10.1007/s11071-011-0029-z.  Google Scholar

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##### References:
 [1] H. Chen and J. Sun, Stability analysis for coupled systems with time delay on networks, Physica A, 391 (2012), 528-534.  doi: 10.1016/j.physa.2011.08.037.  Google Scholar [2] C. Cheng, T. Liao and C. Wang, Exponential synchronization of a class of chaotic neural networks, Chaos Soliton. Fract., 24 (2005), 197-206.  doi: 10.1016/S0960-0779(04)00566-1.  Google Scholar [3] C. Cheng, Robust synchronization of uncertain unified chaotic systems subject to noise and its application to secure communication, Appl. Math. Comput., 219 (2012), 2698-2712.  doi: 10.1016/j.amc.2012.08.101.  Google Scholar [4] Y. Dong and J. Chen, Finite-time outer synchronization between two complex dynamical networks with on-off coupling, Int. J. Model Phys. C, 26 (2015), 1550095, 13 pp.  Google Scholar [5] P. Du and M. Y. Li, Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations, Physica D, 286/287 (2014), 32-42.  doi: 10.1016/j.physd.2014.07.008.  Google Scholar [6] W. Du, J. Zhang, X. An, S. Qin and J. Yu, Outer synchronization between two coupled complex networks and its application in public traffic supernetwork, Discrete Dyn. Nat. Soc. , 2016 (2016), Art. ID 8920764, 8 pp.  Google Scholar [7] R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis, Discrete cont. Dyn-B, 19 (2014), 1355-1372.  doi: 10.3934/dcdsb.2014.19.1355.  Google Scholar [8] Y. Guo, S. Liu and X. Ding, The existence of periodic solutions for coupled Rayleigh system, Neurocomputing, 191 (2016), 398-408.  doi: 10.1016/j.neucom.2016.01.039.  Google Scholar [9] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar [10] H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261-279.  doi: 10.1137/110827028.  Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.  Google Scholar [12] Y. Kao and C. Wang, Global stability analysis for stochastic coupled reaction-diffusion systems on networks, Nonlinear Anal. RWA., 14 (2013), 1457-1465.  doi: 10.1016/j.nonrwa.2012.10.008.  Google Scholar [13] R. Leander, S. Lenhart and V. Protopopescu, Controlling synchrony in a network of Kuramoto oscillators with time-varying coupling, Physica D, 301/302 (2015), 36-47.  doi: 10.1016/j.physd.2015.03.003.  Google Scholar [14] T. Li, B. Rao and Y. Wei, Generalized exact boundary synchronization for a coupled system of wave equation, Discrete cont. Dyn-A, 34 (2014), 2893-2905.   Google Scholar [15] W. Li, S. Liu and D. Xu, The existence of periodic solutions for coupled pantograph Rayleigh system, Math. Methods Appl. Sci., 39 (2016), 1667-1679.  doi: 10.1002/mma.3556.  Google Scholar [16] C. Li, W. Sun and J. Kurths, Synchronization between two coupled complex networks, Phys. Rev. E, 76 (2007), 046204.  doi: 10.1103/PhysRevE.76.046204.  Google Scholar [17] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar [18] W. Li, H. Su and K. Wang, Global stability analysis for stochastic coupled systems on networks, Automatica, 47 (2011), 215-220.  doi: 10.1016/j.automatica.2010.10.041.  Google Scholar [19] B. Lisena, Average criteria for periodic neural networks with delay, Discrete cont. Dyn-B, 19 (2014), 761-773.  doi: 10.3934/dcdsb.2014.19.761.  Google Scholar [20] X. Liu and T. Chen, Boundedness and synchronization of $y$-coupled Lorenz systems with or without controllers, Physica D, 237 (2008), 630-639.  doi: 10.1016/j.physd.2007.10.006.  Google Scholar [21] Y. Lou, W. M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete cont. Dyn-A, 35 (2015), 1589-1607.   Google Scholar [22] K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, T. Am. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar [23] K. Modin and O. Verdier, Integrability of nonholonomically coupled oscillators, Discrete cont. Dyn-A, 34 (2014), 1121-1130.   Google Scholar [24] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA., 13 (2012), 1581-1592.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar [25] S. H. Strogatz, Exploring complex networks, Nature, 140 (2001), 268-276.  doi: 10.1038/35065725.  Google Scholar [26] H. Su, W. Li and K. Wang, Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos, 22 (2012), 033135, 11pp.  Google Scholar [27] R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160.  doi: 10.1137/0201010.  Google Scholar [28] H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mt. J. Math., 24 (1994), 351-380.   Google Scholar [29] J. P. Tseng, Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays, Discrete cont. Dyn-A, 33 (2013), 4693-4729.  doi: 10.3934/dcds.2013.33.4693.  Google Scholar [30] G. Wang, J. Cao and J. Lu, Outer synchronization between two nonidentical networks with circumstance noise, Physica A, 389 (2010), 1480-1488.  doi: 10.1016/j.physa.2009.12.014.  Google Scholar [31] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.  Google Scholar [32] J. Xu, D. Park and J. Jo, Local complexity predicts global synchronization of hierarchically networked oscillators, Chaos, 27 (2017), 073116, 11pp.  Google Scholar [33] C. Zhang, W. Li and K. Wang, Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1698-1709.  doi: 10.1109/TNNLS.2014.2352217.  Google Scholar [34] L. Zhang, Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks, Discrete cont. Dyn-A, 34 (2014), 2405-2450.   Google Scholar [35] S. Zheng, S. Wang, G. Dong and Q. Bi, Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 284-291.  doi: 10.1016/j.cnsns.2010.11.029.  Google Scholar [36] Q. Zhu and J. Cao, pth moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching, Nonlinear Dyn., 67 (2012), 829-845.  doi: 10.1007/s11071-011-0029-z.  Google Scholar
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.
A general digraph $\mathcal{G}$ with 67 vertices and the corresponding condensed digraph $\mathcal{H}$
Digraph $(\mathcal{G},B)$ with 20 vertices and a not strongly connected network.
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,$\tilde{x}_{4+5i} = -60,x_{5+5i} = 70,\tilde{x}_{5+5i} = -70,i = 0,1,2,3)$.
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,$\tilde{y}_{4+5i} = 30,y_{5+5i} = 40,\tilde{y}_{5+5i} = 20,i = 0,1,2,3)$.
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,$e_{5+5i} = -40,\tilde{e}_{5+5i} = 110,i = 0,1,2,3)$.
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).
 $\mathrm{Forms}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$ $\tau_{k}(t)$ $0.5(\sin(t)+1)$ $0.5(\cos(t)+1)$ $0.4(\sin(t)+1)$ $0.4(\cos(t)+1)$ $N_{h}(x_{h})$ $\sin(x_{h})$ $\cos(x_{h})$ $0.5\sin(2x_{h})$ $0.5\cos(2x_{h})$
 $\mathrm{Forms}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$ $\tau_{k}(t)$ $0.5(\sin(t)+1)$ $0.5(\cos(t)+1)$ $0.4(\sin(t)+1)$ $0.4(\cos(t)+1)$ $N_{h}(x_{h})$ $\sin(x_{h})$ $\cos(x_{h})$ $0.5\sin(2x_{h})$ $0.5\cos(2x_{h})$
 $\mathrm{Parameters}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$ $\epsilon_{k}$ $1.8$ $1.9$ $2.0$ $2.1$ $\varepsilon_{k}$ $0.05$ $0.1$ $0.15$ $0.2$
 $\mathrm{Parameters}$ $k\in\{1,\ldots,5\}$ $k\in\{6,\ldots,10\}$ $k\in\{11,\ldots,15\}$ $k\in\{16,\ldots,20\}$ $\epsilon_{k}$ $1.8$ $1.9$ $2.0$ $2.1$ $\varepsilon_{k}$ $0.05$ $0.1$ $0.15$ $0.2$
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