Outer Synchronization of Partially Coupled Dynamical Networks via Pinning Impulsive Controllers

Living reference work entry


This chapter presents an analytical study of outer synchronization of partially coupled dynamical networks via pinning impulsive controller. At first, more realistic drive-response partially coupled networks are established. Then, based on the regrouping method, some efficient and less conservative synchronization criteria are derived and developed in terms of average impulsive interval. The results show that, by impulsively controlling a crucial fraction of nodes in the response network, the outer synchronization can be achieved. Finally, illustrated examples are given to verify the effectiveness of the proposed strategy.


Complex dynamical networks Partial coupling Outer synchronization Pinning impulsive control Average impulsive 


  1. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)MathSciNetCrossRefGoogle Scholar
  2. A.L. Barabási, R. Albert, H. Jeong, Mean-field theory for scale-free random networks. Phys. A Stat. Mech. Appl. 272(1), 173–187 (1999)CrossRefGoogle Scholar
  3. J. Cao, P. Li, W. Wang, Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353(4), 318–325 (2006)CrossRefGoogle Scholar
  4. T. Chen, X. Liu, W. Lu, Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I Regul. Pap. 54(6), 1317–1326 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. W. Chen, Z. Jiang, J. Zhong, X. Lu, On designing decentralized impulsive controllers for synchronization of complex dynamical networks with nonidentical nodes and coupling delays. J. Frankl. Inst. 351(8), 4084–4110 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  6. A.B. Horne, T.C. Hodgman, H.D. Spence, A.R. Dalby, Constructing an enzyme-centric view of metabolism. Bioinformatics 20(13), 2050–2055 (2004)CrossRefGoogle Scholar
  7. A. Hu, Z. Xu, Pinning a complex dynamical network via impulsive control. Phys. Lett. A 374(2), 186–190 (2009)zbMATHCrossRefGoogle Scholar
  8. J. Hu, J. Liang, J. Cao, Synchronization of hybrid-coupled heterogeneous networks: pinning control and impulsive control schemes. J. Frankl. Inst. 351(5), 2600–2622 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  9. C. Hua, C. Ge, X. Guan, Synchronization of chaotic Lur’e systems with time delays using sampled-data control. IEEE Trans. Neural Netw. Learn. Syst. 26(6), 1214–1221 (2015)MathSciNetCrossRefGoogle Scholar
  10. C. Huang, D.W.C. Ho, J. Lu, J. Kurths, Partial synchronization in stochastic dynamical networks with switching communication channels. Chaos Interdisciplinary J. Nonlinear Sci. 22(2), 023108 (2012a)MathSciNetzbMATHCrossRefGoogle Scholar
  11. C. Huang, D.W.C. Ho, J. Lu, Partial-information-based distributed filtering in two-targets tracking sensor network. IEEE Trans. Circuits Syst. I Regul. Pap. 59(4), 820–832 (2012b)MathSciNetCrossRefGoogle Scholar
  12. M. Newmann, The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefGoogle Scholar
  13. M.E.J. Newman, D.J. Watts, Scaling and percolation in the small-world network model. Phys. Rev. E 60(6), 7332–7342 (1999)CrossRefGoogle Scholar
  14. V.L. Krinsky, V.N. Biktashev, I.R. Efimov, Autowave principles for parallel image processing. Phys. D Nonlinear Phenom. 49(1), 247–253 (1991)CrossRefGoogle Scholar
  15. X. Li, X. Wang, G. Chen, Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circuits Syst. I Regul. Pap. 51(10), 2074–2087 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. C. Li, W. Sun, J. Kurths, Synchronization between two coupled complex networks. Phys. Rev. E 76(4), 046204 (2007)Google Scholar
  17. C. Li, S. Wu, G. Feng, X. Liao, Stabilizing effects of impulses in discrete-time delayed neural networks. IEEE Trans. Neural Netw. 22(2), 323–329 (2011)CrossRefGoogle Scholar
  18. L. Li, D.W.C. Ho, J. Lu, A unified approach to practical consensus with quantized data and time delay. IEEE Trans. Circuits Syst I Regul. Pap. 60(10), 2668–2678 (2013)MathSciNetCrossRefGoogle Scholar
  19. X. Liu, Stability results for impulsive differential systems with applications to population growth models. Dyn. Stab. Syst. 9(2), 163–174 (1994)MathSciNetzbMATHGoogle Scholar
  20. B. Liu, W. Lu, T. Chen, Pinning consensus in networks of multiagents via a single impulsive controller. IEEE Trans. Neural Netw. Learn. Syst. 24(7), 1141–1149 (2013)CrossRefGoogle Scholar
  21. J. Lu, J. Cao, Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn. 53(1–2), 107–115 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. W. Lu, T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems. Phys. D Nonlinear Phenom. 213(2), 214–230 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. J. Lu, D.W.C. Ho, Globally exponential synchronization and synchronizability for general dynamical networks. IEEE Trans. Syst. Man Cybern. Part B Regul. Pap. 40(2), 350–361 (2010)CrossRefGoogle Scholar
  24. J. Lu, D.W.C. Ho, L. Wu, Exponential stabilization in switched stochastic dynamical networks. Nonlinearity 22, 889–911 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. J. Lu, D. W. C. Ho, J. Cao, A unified synchronization criterion for impulsive dynamical networks. Automatica 46(7), 1215–1221 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. J. Lu, J. Kurths, J. Cao, N. Mahdavi, C. Huang, Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 285–292 (2012)CrossRefGoogle Scholar
  27. J. Lu, D.W.C. Ho, J. Cao, J. Kurths, Single impulsive controller for globally exponential synchronization of dynamical networks. Nonlinear Anal. Real World Appl. 14(1), 581–593 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  28. J. Lu, J. Zhong, Y. Tang, T. Huang, J. Cao, J. Kurths, Synchronization in output-coupled temporal Boolean networks. Sci. Rep. 4, 6292–6303 (2014)CrossRefGoogle Scholar
  29. J.Q. Lu, C.D. Ding, J.G. Lou, J.D. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. J. Frankl. Inst. 352, 5024–5041 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  30. S.H. Strogatz, Exploring complex networks. Nature 410(6825), 268–276 (2001)zbMATHCrossRefGoogle Scholar
  31. W. Sun, J. Lü, S. Chen, X. Yu, Pinning impulsive control algorithms for complex network. Chaos Interdisciplinary J. Nonlinear Sci. 24(1), 013141 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  32. H. Tang, L. Chen, J. Lu, K.T. Chi, Adaptive synchronization between two complex networks with nonidentical topological structures. Phys. A Stat. Mech. Appl. 387(22), 5623–5630 (2008)CrossRefGoogle Scholar
  33. Y. Tang, W.K. Wong, J.A. Fang, Pinning impulsive synchronization of stochastic delayed coupled networks. Chin. Phys. B 20(4), 040513 (2011)CrossRefGoogle Scholar
  34. Y. Tang, H. Gao, J. Lu, J. Kurths, Pinning distributed synchronization of stochastic dynamical networks: a mixed optimization method. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1804–1815 (2014a)CrossRefGoogle Scholar
  35. Y. Tang, Z. Wang, H. Gao, H. Qiao, J. Kurths, On controllability of neuronal networks with constraints on the average of control gains. IEEE Trans. Cybern. 44(12), 2670–2681 (2014b)CrossRefGoogle Scholar
  36. X.F. Wang, Complex networks: topology, dynamics and synchronization. Int. J. Bifurcation Chaos 12(05), 885–916 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  37. X. Wang, G. Chen, Pinning control of scale-free dynamical networks. Phys. A Stat. Mech. Appl. 310(3), 521–531 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  38. T. Wang, H. Gao, J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Trans. Neural Netw. Learn. Syst. (2015). Google Scholar
  39. D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)zbMATHCrossRefGoogle Scholar
  40. W. Wu, Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling. IEEE Trans. Circuits Syst. Express Briefs 52(5), 282–286 (2005)CrossRefGoogle Scholar
  41. C. Wu, L.O. Chua, Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circuits Syst. I Fund. Theory Appl. 42(8), 430–447 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  42. X. Wu, W. Zheng, J. Zhou, Generalized outer synchronization between complex dynamical networks. Chaos Interdisciplinary J. Nonlinear Sci. 19(1), 013109 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  43. W. Yu, G. Chen, J. Lü, On pinning synchronization of complex dynamical networks. Automatica 45(2), 429–435 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  44. T. Yang, Impulsive systems and control: theory and applications (Nova Science Publishers, Inc., New York, 2001)Google Scholar
  45. Z.C. Yang, D. Xu, Stability analysis of delay neural networks with impulsive effects. IEEE Trans. Circuits Syst. II Express Briefs 52(8), 517–521 (2005)CrossRefGoogle Scholar
  46. X. Yang, J. Cao, J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control. IEEE Trans. Circuits Syst. I Regul. Pap. 59(2), 371–384 (2012)MathSciNetCrossRefGoogle Scholar
  47. X. Yang, J. Cao, J. Lu, Synchronization of randomly coupled neural networks with Markovian jumping and time-delay. IEEE Trans. Circuits Syst I. Regul. Pap. 60(2), 363–376 (2013)MathSciNetCrossRefGoogle Scholar
  48. A. Zheleznyak, L.O. Chua, Coexistence of low-and high-dimensional spatiotemporal chaos in a chain of dissipatively coupled Chua’s circuits. Int. J. Bifurcation Chaos 4(03), 639–674 (1994)zbMATHCrossRefGoogle Scholar
  49. J. Zhou, Q. Wu, Exponential stability of impulsive delayed linear differential equations. IEEE Trans. Circuits Syst. II Express Briefs 56(9), 744–748 (2009)CrossRefGoogle Scholar
  50. W. Zhang, Y. Tang, Q. Miao, W. Du, Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. 24(8), 1316–1326 (2013)CrossRefGoogle Scholar
  51. J. Zhong, J. Lu, Y. Liu, J. Cao, Synchronization in an array of output-coupled Boolean networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 25(12), 2288–2294 (2014)CrossRefGoogle Scholar
  52. Z. Zuo, J. Zhang, Y. Wang, Adaptive fault tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems. IEEE Trans. Ind. Electron. 62(6), 3923–3931 (2015)Google Scholar

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingChina
  2. 2.School of Information EngineeringHuzhou UniversityHuzhouChina

Section editors and affiliations

  • Yang Tang
    • 1
  1. 1.Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of EducationEast China University of Science and TechnologyShanghaiChina

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