Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 523424, 10 pages
http://dx.doi.org/10.1155/2015/523424
Research Article

Exponential Cluster Synchronization of Neural Networks with Proportional Delays

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 15 January 2015; Revised 9 March 2015; Accepted 24 March 2015

Academic Editor: Kun Liu

Copyright © 2015 Nian Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. X. F. Wang, “Complex networks: topology, dynamics and synchronization,” International Journal of Bifurcation and Chaos, vol. 12, no. 5, pp. 885–916, 2002. View at Publisher · View at Google Scholar · View at Scopus
  2. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Scopus
  3. M. Feki, “An adaptive chaos synchronization scheme applied to secure communication,” Chaos, Solitons & Fractals, vol. 18, no. 1, pp. 141–148, 2003. View at Publisher · View at Google Scholar · View at Scopus
  4. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Paperback), Cambridge Nonlinear Science Series, 2003.
  5. P. de Lellis, M. di Bernardo, and F. Garofalo, “Synchronization of complex networks through local adaptive coupling,” Chaos, vol. 18, no. 3, Article ID 037110, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. W. Ren, “Synchronization of coupled harmonic oscillators with local interaction,” Automatica, vol. 44, no. 12, pp. 3195–3200, 2008. View at Publisher · View at Google Scholar · View at Scopus
  7. W. Lu, B. liu, and T. Chen, “Cluster synchronization in networks of coupled non-identical dynamical systems,” Chaos, vol. 20, no. 1, Article ID 013120, 2010. View at Publisher · View at Google Scholar
  8. Y. Wang and J. Cao, “Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 842–851, 2013. View at Publisher · View at Google Scholar · View at Scopus
  9. Z. Ma, Z. Liu, and G. Zhang, “A new method to realize cluster synchronization in connected chaotic networks,” Chaos, vol. 16, no. 2, Article ID 023103, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. W. Wu, W. Zhou, and T. Chen, “Cluster synchronization of linearly coupled complex networks under pinning control,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 4, pp. 829–839, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Zhou, T. Chen, and L. Xiang, “Robust synchronization of delayed neural networks based on adaptive control and parameters identification,” Chaos, Solitons and Fractals, vol. 27, no. 4, pp. 905–913, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. C.-J. Cheng, T.-L. Liao, J.-J. Yan, and C.-C. Hwang, “Exponential synchronization of a class of neural networks with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 36, no. 1, pp. 209–215, 2006. View at Publisher · View at Google Scholar · View at Scopus
  13. J.-S. Lin, M.-L. Hung, J.-J. Yan, and T.-L. Liao, “Decentralized control for synchronization of delayed neural networks subject to dead-zone nonlinearity,” Nonlinear Analysis, Theory, Methods & Applications, vol. 67, no. 6, pp. 1980–1987, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. X. Li and M. Bohner, “Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 643–653, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. J. Lu, D. W. C. Ho, J. Cao, and J. Kurths, “Exponential synchronization of linearly coupled neural networks with impulsive disturbances,” IEEE Transactions on Neural Networks, vol. 22, no. 2, pp. 329–335, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. X. Yang, Z. Wu, and J. Cao, “Finite-time synchronization of complex networks with non-identical discontinuous nodes,” Nonlinear Dynamics, vol. 73, no. 4, pp. 2313–2327, 2013. View at Publisher · View at Google Scholar · View at Scopus
  17. W.-T. Miao, T. Yuan, J.-W. Xiao, and Y.-W. Wang, “Adaptive synchronization of complex dynamical networks with two types of time-varying delays,” in Proceedings of the 8th IEEE International Conference on Control and Automation (ICCA '10), pp. 1584–1588, IEEE, Xiamen, China, June 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Physical Review E, vol. 86, no. 1, part 2, Article ID 016202, 2012. View at Publisher · View at Google Scholar
  19. Y. Chen, M. Hamdi, and D. H. K. Tsang, “Proportional QoS over OBS networks,” in Proceedings of the IEEE Global Telecommunicatins Conference (GLOBECOM '01), pp. 1510–1514, November 2001. View at Scopus
  20. J. Wei, C.-Z. Xu, X. Zhou, and Q. Li, “A robust packet scheduling algorithm for proportional delay differentiation services,” Computer Communications, vol. 29, no. 18, pp. 3679–3690, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Kulkarni, R. Sharma, and I. Mishra, “New QoS routing algorithm for MPLS networks using delay and bandwidth constrainst,” International Journal of Information and Communication Technology Research, vol. 2, no. 3, pp. 285–293, 2012. View at Google Scholar
  22. J. Liu, D. Yuan, S. Ci, and Y. Zhong, “A new QoS routing optimal algorithm in mobile ad hoc networks based on hopfield neural network,” in Advances in Neural Networks—ISNN 2005, vol. 3498 of Lecture Notes in Computer Science, pp. 343–348, Springer, Berlin, Germany, 2005. View at Publisher · View at Google Scholar
  23. R. Gargi, Y. Chaba, and R. B. Patel, “Improving the performance of dynamic source routing protocol by optimization of neural networks,” International Journal of Computer Science Issues, vol. 9, Issue 4, no. 3, 2012. View at Google Scholar
  24. B. van Brunt, J. C. Marshall, and G. C. Wake, “Holomorphic solutions to pantograph type equations with neutral fixed points,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 557–569, 2004. View at Publisher · View at Google Scholar · View at Scopus
  25. Y. Liu, “Asymptotic behaviour of functional-differential equations with proportional time delays,” European Journal of Applied Mathematics, vol. 7, no. 1, pp. 11–30, 1996. View at Google Scholar · View at Scopus
  26. L. Zhou, “Delay-dependent exponential stability of cellular neural networks with multi-proportional delays,” Neural Processing Letters, vol. 38, no. 3, pp. 347–359, 2013. View at Publisher · View at Google Scholar · View at Scopus
  27. L. Zhou, “Dissipativity of a class of cellular neural networks with proportional delays,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1895–1903, 2013. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Iserles and Y. Liu, “On neutral functional–differential equations with proportional delays,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 73–95, 1997. View at Publisher · View at Google Scholar · View at Scopus
  29. H. Brunner, Q. Hu, and Q. Lin, “Geometric meshes in collocation methods for Volterra integral equations with proportional delays,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 783–798, 2001. View at Publisher · View at Google Scholar · View at Scopus
  30. L. Zhou, X. Chen, and Y. Yang, “Asymptotic stability of cellular neural networks with multiple proportional delays,” Applied Mathematics and Computation, vol. 229, no. 5, pp. 457–466, 2014. View at Publisher · View at Google Scholar · View at Scopus
  31. L. Zhou, “Global asymptotic stability of cellular neural networks with proportional delays,” Nonlinear Dynamics, vol. 771, pp. 41–47, 2014. View at Publisher · View at Google Scholar · View at Scopus
  32. J. Cao and L. Li, “Cluster synchronization in an array of hybrid coupled neural networks with delay,” Neural Networks, vol. 22, no. 4, pp. 335–342, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. C.-J. Cheng, T.-L. Liao, J.-J. Yan, and C.-C. Hwang, “Synchronization of neural networks by decentralized feedback control,” Physics Letters A: General, Atomic and Solid State Physics, vol. 338, no. 1–18, pp. 28–35, 2005. View at Publisher · View at Google Scholar · View at Scopus
  34. A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999. View at Publisher · View at Google Scholar · View at Scopus
  35. K. Liu, E. Fridmana, and L. Hetel, “Stability and l2-gain analysis of networked control systems under Round-Robin scheduling: a time-delay approach,” Systems & Control Letters, vol. 61, no. 8, pp. 666–675, 2012. View at Google Scholar
  36. W. P. M. H. Heemels, A. R. Teel, N. van de Wouw, and D. Nešić, “Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1781–1796, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. K. Liu, E. Fridman, and L. Hetel, “Network-based control via a novel analysis of hybrid systems with time-varying delays,” in Proceedings of the IEEE 51st Conference on Decision and Control (CDC '12), pp. 3886–3891, December 2012. View at Publisher · View at Google Scholar · View at Scopus