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Global Synchronization

  • Mattia Frasca
  • Lucia Valentina Gambuzza
  • Arturo Buscarino
  • Luigi Fortuna
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

In this chapter, we discuss the most studied form of synchronization in a network, where all the units asymptotically converge toward the same trajectory. After describing the phenomenon and the measures to characterize it, we illustrate the methods for the study of the stability and present some results on how to control it.

References

  1. 1.
    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Complex networks: structure and dynamics. Phys. Rep. 424(4), 175–308 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Buscarino, L. Fortuna, M. Frasca, Essentials of Nonlinear Circuit Dynamics with MATLAB® and Laboratory Experiments (CRC Press, 2017)Google Scholar
  3. 3.
    F. Chen, Z. Chen, L. Xiang, Z. Liu, Z. Yuan, Reaching a consensus via pinning control. Automatica 45(5), 1215–1220 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Chen, Z. Duan, Network synchronizability analysis: a graph-theoretic approach. Chaos: Interdiscip. J. Nonlinear Sci. 18(3), 037102 (2008)zbMATHGoogle Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    P. DeLellis, M. di Bernardo, M. Porfiri, Pinning control of complex networks via edge snapping. Chaos: Interdiscipl. J. Nonlinear Sci. 21(3), 033119 (2011a)zbMATHGoogle Scholar
  7. 7.
    P. DeLellis, M. di Bernardo, G. Russo, On quad, lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circuits Syst. I: Regul. Pap. 58(3), 576–583 (2011b)Google Scholar
  8. 8.
    P. DeLellis, F. Garofalo, F.L. Iudice, The partial pinning control strategy for large complex networks. Automatica 89, 111–116 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    O.L. de Weck, MIT Strategic Engineering Research Group, Matlab tools for network analysis, http://strategic.mit.edu/downloads.php?page=matlab_networks, 2014. Accessed 3 Jan 2018
  10. 10.
    Z. Duan, G. Chen, G, L. Huang, Complex network synchronizability: analysis and control. Phys. Rev. E 76(5), 056103 (2007)CrossRefGoogle Scholar
  11. 11.
    Z. Duan, C. Liu, G. Chen, Network synchronizability analysis: the theory of subgraphs and complementary graphs. Physica D: Nonlinear Phenom. 237(7), 1006–1012 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Frasca, A. Buscarino, A. Rizzo, L. Fortuna, S. Boccaletti, Synchronization of moving chaotic agents. Phys. Rev. Lett. 100(4), 044102 (2008)CrossRefGoogle Scholar
  13. 13.
    L. Huang, Q. Chen, Y.C. Lai, L.M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E 80(3), 036204 (2009)CrossRefGoogle Scholar
  14. 14.
    X. Li, G. Chen, Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 50(11), 1381–1390 (2003)Google Scholar
  15. 15.
    Z. Li, G. Chen, Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. Circuits Syst. II: Express. Briefs 53(1), 28–33 (2006)Google Scholar
  16. 16.
    J. Lu, X. Yu, G. Chen, D. Cheng, Characterizing the synchronizability of small-world dynamical networks. IEEE Trans. Circuits Syst. I: Regul. Pap. 51(4), 787–796 (2004)Google Scholar
  17. 17.
    W. Lu, T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems. Physica D: Nonlinear Phenom. 213(2), 214–230 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    W. Lu, X. Li, Z. Rong, Global stabilization of complex networks with digraph topologies via a local pinning algorithm. Automatica 46(1), 116–121 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R. N. Madan, Chua’s Circuit: A Paradigm for Chaos, vol. 1. (World Scientific, 1993)Google Scholar
  20. 20.
    P.J. Menck, J. Heitzig, N. Marwan, J. Kurths, How basin stability complements the linear-stability paradigm. Nat. Phys. 9(2), 89–92 (2013)CrossRefGoogle Scholar
  21. 21.
    T. Nishikawa, A.E. Motter, Synchronization is optimal in nondiagonalizable networks. Phys. Rev. E 73(6), 065106 (2006)CrossRefGoogle Scholar
  22. 22.
    R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefzbMATHGoogle Scholar
  23. 23.
    L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109 (1998)CrossRefGoogle Scholar
  24. 24.
    M. Porfiri, M. Di Bernardo, Criteria for global pinning-controllability of complex networks. Automatica 44(12), 3100–3106 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)CrossRefzbMATHGoogle Scholar
  26. 26.
    Q. Song, J. Cao, On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans. Circuits Syst. I: Regul. Pap. 57(3), 672–680 (2010)Google Scholar
  27. 27.
    D.J. Stilwell, E.M. Bollt, D.G. Roberson, Sufficient conditions for fast switching synchronization in time-varying network topologies. SIAM J. Appl. Dyn. Syst. 5(1), 140–156 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    J. Sun, E.M. Bollt, T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems. EPL (Europhys. Lett.) 85(6), 60011 (2009)Google Scholar
  29. 29.
    X.F. Wang, G. Chen, Pinning control of scale-free dynamical networks. Physica A: Stat. Mech. Appl. 310(3), 521–531 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    W. Yu, G. Chen, J. Lü, On pinning synchronization of complex dynamical networks. Automatica 45(2), 429–435 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Mattia Frasca
    • 1
  • Lucia Valentina Gambuzza
    • 1
  • Arturo Buscarino
    • 1
  • Luigi Fortuna
    • 1
  1. 1.Department of Electrical, Electronic and Computer EngineeringUniversity of CataniaCataniaItaly

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