基于Lyapunov稳定性理论,研究了不确定复杂网络的广义矩阵投影同步,每个网络的拓扑结构不恒等且维数也不同。在节点参数未知的情况下,通过设计自适应控制器实现了两个不同的变时滞复杂网络的广义矩阵投影同步,未知参数可以辨识。此外,对于未知的驱动网络及给定的广义矩阵,构造了实现广义矩阵投影同步的响应网络,不仅可以对该驱动网络进行同步控制达到预期效果,而且能对未知参数进行辨识确定网络结构。最后数值仿真验证了方法的有效性和可行性。
The generalized matrix projective synchronization of uncertain complex networks is investigated in this paper based on the Lyapunov stability theory, where each network may have non-identical topological structure and different dimensions. When parameters of nodes are uncertain, two different complex networks with time-varying delay coupling can realize GMPS by constructing an adaptive controller. In addition, for the unknown drive network and a given generalized matrix, the response network is constructed to realize GMPS. It can not only control the network to achieve the synchronization but also identify the unknown parameters to affirm the structure of the network. Finally, illustrative examples are presented to demonstrate the application of the theoretical results.
[1] Watts D J,Stogatz S H. Collective dynamics of 'small-world' networks[J]. Nature, 1998, 393 (6684): 440-442.
[2] Barabasi A L,Albert R.Emergence of Scaling in Random Networks[J]. Science, 1998, 286 (5439): 509-520.
[3] Pecora L M, Carroll T L. Synchronization in chaotic system[J]. Physical Review Letters, 1990, 64 (8): 821-824.
[4] Agiza H N. Chaos synchronization of Lü dynamical system[J]. Nonlinear Analysis, 2004, 58 (1/2): 11-20.
[5] Pikovsky A S,Rosenblum M G,Osipov G V. Phase synchronization of chaotic oscillators by external driving[J]. Physica D: Nonlinear Phenomena, 1997, 104(3/4): 219-238.
[6] Li C D. Liao X F,Wong K W. Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication[J]. Physica D: Nonlinear Phenomen, 2004, 194(3/4): 187-202.
[7] Rulkov N F.Sushchik M M,Tsimring L S,et al. Gneralized synchronization of chaos in directionally coupled chaotic systems[J]. Phys Rev E, 1995, 51(4): 980-994.
[8] Yan J P, Li C P. Generalized projective synchronization of a unified chaotic system[J]. Chaos Solitons and Fractals, 2005, 26(4): 1119-1124.
[9] Li G H. Modified projective synchronization of chaotic system[J]. Chaos Solitons and Fractals, 2007, 32 (5): 1786-1790.
[10] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems[J].Physical Review letters, 1999, 82 (15): 3042-3045.
[11] Li C P, Yan J P. Generalized projective synchronization of chaos: The cascade synchronization approach[J].Chaos Solitons and Fractals, 2006, 30 (1): 140-146.
[12] Li G H. Generalized projective synchronization of two chaotic systems by using active control[J].Chaos Solitons and Fractals, 2006, 30 (1): 77-82.
[13] Li Z G, Xu D L. Stability criterion for projective synchronization in three-dimensional chaotic systems[J]. Phys Letter A, 2001, 282 (3): 175-179.
[14] 柴秀丽,武相军.一类参数未知超混沌系统的广义函数投影滞后同步[J].计算机应用,2013, 33(3): 734-738.
Cai Xiuli, Wu Xiangjun, Generalized function projective lag synchronization of a class of hyperchaotic systems with fully uncertain parameters[J]. Journal of Computer Applications, 2013, 33(3): 734-738.
[15] Dai H. Jia L X. Zhang Y B. Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions[J]. Chin Phys B, 2012, 21 (12): 141-152.
[16] Zhou J, Lu J A. Topology identification of weighted complex dynamical network[J]. Physica A, 2007, 386 (1): 481-491.
[17] Li K, Lai C H, Adaptive-impulsive synchronization of uncertain complex dynamical networks[J].Physics Letters A, 2008,372 (10): 1601-1606.
[18] Xu Y H, Zhou W N, Fang J A, et al. Topology identification and adaptive synchronization of uncertain complex networks with non-derivative and derivative coupling[J].Journal of the Franklin Institute, 2010, 347 (8): 1566-1576.
[19] Xu Y H. Zhou W N. Fang J A. Sun W. Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions[J]. Common Nonlinear Sci Numer Simulat, 2011, 16(8): 3337-3343.
[20] Wu X J,Lu H. Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes[C]. Commun Nonlinear Sci Numer Simulat, 2012, 17 (7): 3005-3021.
[21] 李德奎,张建刚.时滞和非时滞耦合的驱动响应动态网络的函数投影同步[J].太原理工大学学报,2013,44(2):162-167.
Li Dekui, Zhang Jiangang. Function projection synchronization of drive-response dynamical networks with non-delayed and delayed coupling[J]. Journal of Taiyuan University of Technology, 2013, 44(2):162-167.
[22] 卞秋香,姚洪兴.复杂网络的线性广义同步[J].系统工程理论与实践,2011,31(7):1334-1340.
Bain Qiuxiang, Yao Hongxing. Linear generalized synchronization of complex networks[J]. Systems Engineering-Theory and Practice, 2011,31(7):1334-1340.
