Stabilization of delayed networks with different nodes

Abstract

In this paper, the control of complex delayed networks with different nodes is proposed. Firstly, the stabilization of coupled networks with time delay is investigated. By constructing a Lyapunov function, a linear feedback controller design procedure for the networks is converted to the problem of solving a set of linear matrix inequalities. Then the results are extended to networks with both delayed dynamical nodes and delayed couplings. It is shown that the stabilization of complex networks is determined by the dynamics of each uncoupled node, coupling matrix and feedback gain matrix of networks. Two examples are simulated. In the first example, a network with 10 nodes consisting of Lorenz systems and systems proposed by Zhang in 2009 is given. It is found that the network states are divergent without control, and convergent under designed linear feedback controllers. In the second example, a larger network with 100 nodes consisting of delayed Chen systems and delayed Lorenz systems is given. The proposed method is also effective for large scale networks.

Keywords

complex network / time delay / linear feedback / linear matrix inequalities

References

[1] Strogatz S. Exploring complex networks[J]. Nature, 2001, 410(6825): 268-276.

[2] Wang X., Chen Guanrong. Synchronization in scalefree dynamical networks: Robustness and fragility[J]. IEEE Transactions on Circuits and Systems I, 2002, 49(1): 54-62.

[3] Lü J., Yu X., Chen Guanrong. Chaos synchronization of general complex dynamical networks[J]. Physica A, 2004, 334(1/2): 281-302.

[4] Li Z., Feng G., Hill David. Controlling complex dynamical networks with coupling delays to a desired orbit [J]. Physics Letters A, 2006, 359(1): 42-46.

[5] Li C., Chen Guanrong. Synchronization in general complex dynamical networks with coupling delays [J]. Physica A, 2004, 343, 263-278.

[6] Zhou J., Chen Tianping. Synchronization in general complex delayed dynamical networks[J]. IEEE Transactions on Circuits and Systems I, 2006, 53(3): 733-744.

[7] Gao H., Lam J., Chen Guanrong. New criteria for synchronization stability of general complex dynamical networks with coupling delays[J]. Physics Letters A, 2006, 360(2): 263-273.

[8] Li P., Yi Z., Zhang Lei. Global synchronization of a class of delayed complex networks[J]. Chaos, Solitons and Fractals, 2006, 30(4): 903-908.

[9] Liu B., Teo K. L., Liu Xinzhi. Global synchronization of dynamical networks with coupling time delays[J]. Physics Letters A, 2007, 368(1/2): 53-63.

[10] Xiang L., Chen Z., Liu Z., et al. Pinning control of complex dynamical networks with heterogeneous delays[J]. Computers and Mathematics with Applications, 2008, 56(5): 1423-1433.

[11] Chen G., Zhou J., Liu Zengrong. Global synchronization of coupled delayed neural networks and applications to chaotic CNN models[J]. International Journal of Bifurcation and Chaos, 2004, 14(7): 2229-2240.

[12] Lu W., Chen Tianping. Synchronization of coupled connected neural networks with delays[J]. IEEE Transactions on Circuits and Systems I, 2004, 51(12): 2491-2503.

[13] Cai S., Zhou J., Xiang L., et al. Robust impulsive synchronization of complex delayed dynamical networks [J]. Physics Letters A, 2008, 372(30): 4990-4995.

[14] Wang Q., Duan Z., Chen G., et al. Synchronization in a class of weighted complex networks with coupling delays[J]. Physica A, 2008, 387(22): 5616-5622.

[15] Wu J., Jiao Licheng. Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers[J]. Physica A, 2008, 387(8/9): 2111-2119.

[16] Zhang Q., Lu Jun’an. Impulsively control complex networks with different dynamical nodes to its trivial equilibrium [J]. Computers and Mathematics with Applications, 2009, 57(7): 1073-1079.

[17] Boyd S., Ghaoui L. E. I., Feron E., et al. Linear Matrix Inequalities in System and Control Theory[M]. 1994, Philadelphia: Society for Industrial and Applied Mathematics.

[18] Zhang J., Tang Wansheng. Analysis and control for a new chaotic system via piecewise linear feedback [J]. Chaos, Solitons and Fractals, 2009, 42(4): 2181-2190.

[19] Song Y., Wei Junjie. Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos[J]. Chaos, Solitons and Fractals, 2004, 22(1): 75-91.

[20] Ren H., Liu D., Han Chongzhao. Anti-control of chaos via direct time delay feedback [J]. Acta Physica Sinica, 2006, 55(6): 2694-2701.

[21] Barabási A. L., Albert R. Emergence of scaling in random networks[J]. Science, 1999, 286(5439): 509-512.