Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling

In this paper, we discuss exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling. Sufficient conditions for both delay-independent and delay-dependent global exponential synchronization are obtained based on Lyapunov functional method. It is shown that, for some coupled chaotic systems and nonlinear coupling functions, if the coupling delay is less than a positive threshold, then the coupled network will be synchronized. On the other hand, with the increase of coupling delay, the synchronizability of the coupled network will be restrained, even eventually de-synchronized. Finally, an example is provided to demonstrate the effectiveness of the proposed results.

[1]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[2]  Guanrong Chen,et al.  Novel criteria of synchronization stability in complex networks with coupling delays , 2007 .

[3]  Tianping Chen,et al.  Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix , 2008 .

[4]  Tianping Chen,et al.  Exponential Synchronization of Nonlinear Coupled Dynamical Networks , 2007, Int. J. Bifurc. Chaos.

[5]  Leon O. Chua,et al.  Reality of chaos in the double scroll circuit: A computer-assisted proof , 1988 .

[6]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[7]  R. L. Viana,et al.  Spatial correlations and synchronization in coupled map lattices with long-range interactions , 2004 .

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

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

[10]  Tianping Chen,et al.  Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay , 2006 .

[11]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[12]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[13]  M. Newman,et al.  Renormalization Group Analysis of the Small-World Network Model , 1999, cond-mat/9903357.

[14]  P. Hartman.  Ordinary Differential Equations , 1965 .

[15]  Chai Wah Wu,et al.  Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[17]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[18]  N. Rulkov,et al.  Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds. , 1997, Chaos.

[19]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[20]  Xiaofeng Liao,et al.  Synchronization in coupled map lattices with small-world delayed interactions , 2004 .

[21]  Jitao Sun,et al.  Robust synchronization of coupled delayed neural networks under general impulsive control , 2009 .

[22]  Chunguang Li,et al.  Synchronization in general complex dynamical networks with coupling delays , 2004 .

[23]  R. Schwarzenberger.  ORDINARY DIFFERENTIAL EQUATIONS , 1982 .

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

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

[26]  Xiao Fan Wang,et al.  Synchronization in Small-World Dynamical Networks , 2002, Int. J. Bifurc. Chaos.

[27]  Tianping Chen,et al.  Synchronization in general complex delayed dynamical networks , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[28]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[29]  L. Chua,et al.  Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication , 1997 .

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

[31]  Pedro G. Lind,et al.  Impact of bistability in the synchronization of chaotic maps with delayed coupling and complex topologies , 2006 .

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

[33]  Tianping Chen,et al.  Synchronization of coupled connected neural networks with delays , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[34]  B. Bollobás.  The evolution of random graphs , 1984 .