##
**The effect of control strength on lag synchronization of nonlinear coupled complex networks.**
*(English)*
Zbl 1246.93082

Summary: We investigate the lag synchronization of nonlinear coupled complex networks using methods that are based on pinning control, where the weight configuration matrix is not necessarily symmetric or irreducible. We change the control strength into a parameter concerning time \(t\), by using the Lyapunov direct method, some sufficient conditions of lag synchronization are obtained. To validate the proposed method, numerical simulation examples are provided to verify the correctness and effectiveness of the proposed scheme.

### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

PDF
BibTeX
XML
Cite

\textit{S. Wang} and \textit{H. Yao}, Abstr. Appl. Anal. 2012, Article ID 810364, 11 p. (2012; Zbl 1246.93082)

Full Text:
DOI

### References:

[1] | G. M. Mahmoud and E. E. Mahmoud, “Complete synchronization of chaotic complex nonlinear systems with uncertain parameters,” Nonlinear Dynamics, vol. 62, no. 4, pp. 875-882, 2010. · Zbl 1215.93114 |

[2] | S. Xu and Y. Yang, “Global asymptotical stability and generalized synchronization of phase synchronous dynamical networks,” Nonlinear Dynamics, vol. 59, no. 3, pp. 485-496, 2010. · Zbl 1183.70039 |

[3] | J. L. Wang and H. N. Wu, “Local and global exponential output synchronization of complex delayed dynamical networks,” Nonlinear Dynamics, vol. 67, no. 1, pp. 497-504, 2012. · Zbl 1242.93008 |

[4] | Y. C. Hung, Y. T. Huang, M. C. Ho, and C. K. Hu, “Paths to globally generalized synchronization in scale-free networks,” Physical Review E, vol. 77, no. 1, Article ID 016202, 8 pages, 2008. |

[5] | H. N. Agiza, “Chaos synchronization of Lü dynamical system,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 11-20, 2004. · Zbl 1057.34042 |

[6] | N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, “Generalized synchronization of chaos in directionally coupled chaotic systems,” Physical Review E, vol. 51, no. 2, pp. 980-994, 1995. |

[7] | M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Physical Review Letters, vol. 76, no. 11, pp. 1804-1807, 1996. |

[8] | L. P. Zhang, H. B. Jiang, and Q. S. Bi, “Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems,” Nonlinear Dynamics, vol. 59, no. 4, pp. 529-534, 2010. · Zbl 1189.93085 |

[9] | Z.-L. Wang, “Projective synchronization of hyperchaotic Lü system and Liu system,” Nonlinear Dynamics, vol. 59, no. 3, pp. 455-462, 2010. · Zbl 1183.70055 |

[10] | Q. J. Zhang and J. C. Zhao, “Projective and lag synchronization between general complex networks via impulsive control,” Nonlinear Dynamics, vol. 67, no. 4, pp. 2519-2525, 2012. · Zbl 1243.93011 |

[11] | Q. Han, C. D. Li, and J. J. Huang, “Anticipating synchronization of chaotic systems with time delay and parameter mismatch,” Chaos, vol. 19, no. 1, Article ID 013104, 2009. · Zbl 1311.34112 |

[12] | L. Y. Cao and Y. C. Lai, “Anti-phase synchronization in chaotic systems,” Physical Review E, vol. 58, pp. C382-C386, 1998. |

[13] | J. Y. Wang, J. W. Feng, C. Xu, and Y. Zhao, “Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix,” Nonlinear Dynamics, vol. 67, no. 2, pp. 1635-1646, 2012. · Zbl 1242.93009 |

[14] | I. Belykh, V. Belykh, K. Nevidin, and M. Hasler, “Persistent clusters in lattices of coupled nonidentical chaotic systems,” Chaos, vol. 13, no. 1, pp. 165-178, 2003. · Zbl 1080.37525 |

[15] | C. F. Feng, “Projective synchronization between two different time-delayed chaotic systems using active control approach,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 453-459, 2010. · Zbl 1211.93064 |

[16] | W. L. Guo, “Lag synchronization of complex networks via pinning control,” Nonlinear Analysis, vol. 12, no. 5, pp. 2579-2585, 2011, http://www.elsevier.com/locate/nonrwa. · Zbl 1223.93057 |

[17] | E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, “Lag synchronization in time-delayed systems,” Physics Letters A, vol. 292, no. 6, pp. 320-324, 2002. · Zbl 0979.37022 |

[18] | Y. Yang and J. Cao, “Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects,” Physica A, vol. 386, no. 1, pp. 492-502, 2007. |

[19] | C. Li, X. Liao, and K. Wong, “Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication,” Physica D, vol. 194, no. 3-4, pp. 187-202, 2004. · Zbl 1059.93118 |

[20] | Z.-L. Wang and X.-R. Shi, “Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 1091-1097, 2009. · Zbl 1205.37025 |

[21] | J. Zhou, T. Chen, and L. Xiang, “Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication,” Circuits, Systems, and Signal Processing, vol. 24, no. 5, pp. 599-613, 2005. · Zbl 1102.94010 |

[22] | L. P. Wang, Z. T. Yuan, X. H. Chen, and Z. F. Zhou, “Lag synchronization of chaotic systems with parameter mismatches,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 987-992, 2011. · Zbl 1221.37226 |

[23] | Y. Sun and J. Cao, “Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation,” Physics Letters A, vol. 364, no. 3-4, pp. 277-285, 2007. · Zbl 1203.93110 |

[24] | W. Yu and J. Cao, “Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Physica A, vol. 375, no. 2, pp. 467-482, 2007. |

[25] | W. L. Guo, F. Austin, S. H. Chen, and W. Sun, “Pinning synchronization of the complex networks with non-delayed and delayed coupling,” Physics Letters A, vol. 373, no. 17, pp. 1565-1572, 2009. · Zbl 1228.05266 |

[26] | X. F. Wang and G. Chen, “Pinning control of scale-free dynamical networks,” Physica A, vol. 310, no. 3-4, pp. 521-531, 2002. · Zbl 0995.90008 |

[27] | T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Transactions on Circuits and Systems. I, vol. 54, no. 6, pp. 1317-1326, 2007. · Zbl 1374.93297 |

[28] | W. W. Yu, G. R. Chen, and J. H. Lü, “On pinning synchronization of complex dynamical networks,” Automatica A, vol. 45, no. 2, pp. 429-435, 2009. · Zbl 1158.93308 |

[29] | X. Zhou, H. Feng, and S. Chen, “The effect of control strength on the synchronization in pinning control questions,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2014-2018, 2011. · Zbl 1219.93086 |

[30] | W. Yu and J. Cao, “Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Physica A, vol. 375, no. 2, pp. 467-482, 2007. |

[31] | W. Lu and T. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Physica D, vol. 213, no. 2, pp. 214-230, 2006. · Zbl 1105.34031 |

[32] | W. Guo, F. Austin, S. Chen, and W. Sun, “Pinning synchronization of the complex networks with non-delayed and delayed coupling,” Physics Letters A, vol. 373, no. 17, pp. 1565-1572, 2009. · Zbl 1228.05266 |

[33] | C. W. Wu, “Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 5, pp. 282-286, 2005. |

[34] | H. Liu, J. Chen, J. A. Lu, and M. Cao, “Generalized synchronization in complex dynamical networks via adaptive couplings,” Physica A, vol. 389, no. 8, pp. 1759-1770, 2010. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.