Globally Exponential Synchronization and Synchronizability for General Dynamical Networks

The globally exponential synchronization problem for general dynamical networks is considered in this paper. One quantity will be distilled from the coupling matrix to characterize the synchronizability of the corresponding dynamical networks. The calculation of such a quantity is very convenient even for large-scale networks. The network topology is assumed to be directed and weakly connected, which implies that the coupling configuration matrix can be asymmetric, weighted, or reducible. This assumption is more consistent with the realistic network in practice than the constraint of symmetry and irreducibility. By using the Lyapunov functional method and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization of general dynamical networks. In addition, numerical examples, including small-world and scale-free networks, are given to demonstrate the theoretical results. It will be shown that our criteria are available for large-scale dynamical networks.

[1]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[2]  Jinde Cao,et al.  Global Synchronization in an Array of Delayed Neural Networks With Hybrid Coupling , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[3]  Jinde Cao,et al.  Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. , 2005, Chaos.

[4]  Jinde Cao,et al.  Topology influences performance in the associative memory neural networks , 2006 .

[5]  Zidong Wang,et al.  Synchronization and State Estimation for Discrete-Time Complex Networks With Distributed Delays , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[6]  C. Wu.  Synchronization in networks of nonlinear dynamical systems coupled via a directed graph , 2005 .

[7]  Guanrong Chen,et al.  Chaos synchronization of general complex dynamical networks , 2004 .

[8]  Daizhan Cheng,et al.  Characterizing the synchronizability of small-world dynamical networks , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[9]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[10]  Yurong Liu,et al.  Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays , 2006 .

[11]  Tianping Chen,et al.  Pinning Complex Networks by a Single Controller , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  M. Gilli.  Strange attractors in delayed cellular neural networks , 1993 .

[13]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[14]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[16]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[17]  J. Liang,et al.  Robust Synchronization of an Array of Coupled Stochastic Discrete-Time Delayed Neural Networks , 2008, IEEE Transactions on Neural Networks.

[18]  Huijun Gao,et al.  New Delay-Dependent Exponential Stability for Neural Networks With Time Delay , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[19]  Vadim N. Biktashev,et al.  Autowave principles for parallel image processing , 1991 .

[20]  Wei Chen,et al.  Network-based H∞ output tracking control for systems with time-varying bounded delay , 2011, Proceedings 2011 International Conference on Transportation, Mechanical, and Electrical Engineering (TMEE).

[21]  Jinde Cao,et al.  Adaptive synchronization of neural networks with or without time-varying delay. , 2006, Chaos.

[22]  Frank C. Hoppensteadt,et al.  Pattern recognition via synchronization in phase-locked loop neural networks , 2000, IEEE Trans. Neural Networks Learn. Syst..

[23]  Jinde Cao,et al.  Global synchronization in arrays of delayed neural networks with constant and delayed coupling , 2006 .

[24]  Guanrong Chen,et al.  Global Synchronization of Coupled Delayed Neural Networks and Applications to Chaotic CNN Models , 2004, Int. J. Bifurc. Chaos.

[25]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[26]  Zidong Wang,et al.  A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances , 2008 .

[27]  Tianping Chen,et al.  New approach to synchronization analysis of linearly coupled ordinary differential systems , 2006 .

[28]  Huijun Gao,et al.  Network-Based ${{\cal H}}_{\!\!\!\infty }$ Output Tracking Control , 2008, IEEE Transactions on Automatic Control.

[29]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[30]  Zhi Li,et al.  Exponential stability of synchronization in asymmetrically coupled dynamical networks. , 2008, Chaos.

[31]  Jinde Cao,et al.  Adaptive synchronization in tree-like dynamical networks , 2007 .

[32]  Jinde Cao,et al.  Global Synchronization of Linearly Hybrid Coupled Networks with Time-Varying Delay , 2008, SIAM J. Appl. Dyn. Syst..

[33]  Yu Zhao,et al.  Further improvement on synchronization stability of complex networks with coupling delays , 2008, Int. J. Comput. Math..

[34]  Huijun Gao,et al.  Stability analysis for continuous systems with two additive time-varying delay components , 2007, Syst. Control. Lett..

[35]  Xiao Fan Wang,et al.  Synchronization in scale-free dynamical networks: robustness and fragility , 2001, cond-mat/0105014.

[36]  Zidong Wang,et al.  Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances , 2009 .

[37]  Z. Duan,et al.  Network synchronizability analysis: a graph-theoretic approach. , 2008, Chaos.

[38]  Guanrong Chen,et al.  New criteria for synchronization stability of general complex dynamical networks with coupling delays , 2006 .

[39]  Roy,et al.  Communication with chaotic lasers , 1998, Science.

[40]  Zidong Wang,et al.  Global Synchronization Control of General Delayed Discrete-Time Networks With Stochastic Coupling and Disturbances , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[41]  Mao-Yin Chen,et al.  Some Simple Synchronization Criteria for Complex Dynamical Networks , 2006, IEEE Transactions on Circuits and Systems II: Express Briefs.

[42]  D. Ho,et al.  Local and global synchronization in general complex dynamical networks with delay coupling , 2008 .