Synchronization in networks of chaotic systems with time-delay coupling.

In this paper, we consider synchronization of N identical nonlinear systems unidirectionally or bidirectionally coupled with time delay. First we show, using the small-gain theorem, that trajectories of coupled strictly semi-passive systems converge to a bounded region. Next, we consider the network structure under which the synchronization error dynamics has a trivial solution at zero and derive a necessary condition for synchronization with respect to the network structure. Using these facts, we then derive sufficient conditions for synchronization of the systems in terms of linear matrix inequalities via the Lyapunov-Krasovskii functional approach. The obtained results are illustrated on networks of Lorentz systems with coupling delay.

[1]  Wei Xing Zheng,et al.  Global chaos synchronization with channel time-delay , 2004 .

[2]  H Henk Nijmeijer,et al.  Synchronization of Mechanical Systems , 2003 .

[3]  J. Suykens,et al.  Absolute stability theory and master-slave synchronization , 1997 .

[4]  Henk Nijmeijer,et al.  An observer looks at synchronization , 1997 .

[5]  H. Nijmeijer,et al.  Partial synchronization: from symmetry towards stability , 2002 .

[6]  M. G. Earl,et al.  Synchronization in oscillator networks with delayed coupling: a stability criterion. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Johan A. K. Suykens,et al.  Master-Slave Synchronization of Lur'e Systems with Time-Delay , 2001, Int. J. Bifurc. Chaos.

[8]  Alberto Isidori,et al.  Nonlinear Control Systems II , 1999 .

[9]  Yonghui Sun,et al.  Synchronization in an array of linearly stochastically coupled networks with time delays , 2007 .

[10]  N. Buric,et al.  Synchronization of hyperchaotic systems with delayed bidirectional coupling. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[12]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[13]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[14]  Nebojša Vasović,et al.  Excitable systems with internal and coupling delays , 2008 .

[15]  Shengyuan Xu,et al.  Improved delay-dependent stability criteria for time-delay systems , 2005, IEEE Transactions on Automatic Control.

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

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

[18]  Licheng Jiao,et al.  Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers , 2008 .

[19]  Henk Nijmeijer,et al.  Prediction of chaotic behavior , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[20]  Henk Nijmeijer,et al.  Anticipating synchronization of chaotic Lur'e systems. , 2007, Chaos.

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

[22]  Bernhard Maschke,et al.  Dissipative Systems Analysis and Control , 2000 .

[23]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[24]  Chai Wah Wu,et al.  Synchronization in Coupled Chaotic Circuits and Systems , 2002 .

[25]  S. Strogatz,et al.  Time Delay in the Kuramoto Model of Coupled Oscillators , 1998, chao-dyn/9807030.

[26]  S H Strogatz,et al.  Coupled oscillators and biological synchronization. , 1993, Scientific American.

[27]  L. Chua,et al.  A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS , 1994 .

[28]  El-Kebir Boukas,et al.  Deterministic and Stochastic Time-Delay Systems , 2002 .

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

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