Stability conditions for synchronization of networks with mixed couplings by linear stability analysis

  • Wu Rui-xin  (武瑞馨)
  • Zhang Hai-feng  (张海峰)
  • Fu Xin-chu  (傅新楚)
Article
Received:
Revised:
  • 15 Downloads

Abstract

This paper studies some special networks structured with self-organized and driven behavior that coexist in a cluster, moreover, the clusters have dominant intra-cluster and inter-cluster couplings. It is called mixed-system (M-S) here. For this study linear stability analysis was used, and stability conditions for the synchronized state were determined. For the coupling function g(x), the stability state of the network was discussed in two different cases: the linear case with g(x) = x and the nonlinear case with g(x) = f(x). Furthermore, the condition for the emergence of chaos in the networks was given.

Keywords

synchronization mixed-system (M-S) local stability emergence of chaos 

2000 Mathematics Subject Classification

93D99 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Watts D J, Strogatz S H. Collective dynamics of small-world networks [J]. Nature, 1998, 393: 440.CrossRefGoogle Scholar
  2. [2]
    Pandit S A, Amritkar R E. Characterization and control of small-world networks [J]. Physical Review E, 1999, 60: 1119–1122.CrossRefGoogle Scholar
  3. [3]
    Strogatz S H. Exploring complex networks [J]. Nature, 2001, 410: 268–276.CrossRefGoogle Scholar
  4. [4]
    Wu C W. Synchronization in Coupled Chaotic Circuits and System [M]. Singapore: World Scientific, 2002.Google Scholar
  5. [5]
    Wang X, Chen G. Synchronization in small-world dynamical networks [J]. International Journal of Bifurcation and Chaos, 2002, 12: 187–192.CrossRefGoogle Scholar
  6. [6]
    Wang X, Chen G. Synchroization in scale-free dynamical networks: robustness and fragility [J]. IEEE Transactions on Circuits and Systems-I, 2002, 49: 54–62.CrossRefGoogle Scholar
  7. [7]
    Li X, Chen G. Global synchronization and asymptotic stability of complex dynamical networks [J]. IEEE Transactions on Circuits and Systems-II, 2006, 53: 28–33.CrossRefGoogle Scholar
  8. [8]
    Lü J H, Yu X, Chen G. Chaos synchronization of general complex dynamical networks [J]. Physica A, 2004, 334(1–2): 281–302.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Jalan S, Amritkar R E, Hu C K. Synchronized clusters in coupled map networks: stability analysis [J]. Physical Review E, 2005, 72: 016212.Google Scholar
  10. [10]
    Amritkar R E, Jalan S. Self-organized and driven phase synchronization in coupled maps networks [J]. Physica A, 2003, 321: 220–225.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Jalan S, Amritkar R E. Synchronized clusters in coupled map networks: self-organized and driven phase synchronization [J]. Physica Review E, 2005, 72: 016211, and references therein.Google Scholar
  12. [12]
    Jalan S, Amritkar R E. Self-organized and driven phase synchronization in coupled maps [J]. Physical Rewiew Letters, 2003, 90: 014101.Google Scholar
  13. [13]
    Pikovsky A, Rosenblum M, Kurth J. Synchronization: A Universal Concept in Nonlinear Dynamics [M]. Cambridge: Cambridge University Press, 2001.Google Scholar
  14. [14]
    Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos [M]. New York: Springer-Verlag, 1990.Google Scholar
  15. [15]
    Li T Y, Yorke J A. Period three implies chaos [J]. American Mathematical Monthly, 1975, 82: 985–992.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Sharkovskii A N. Coexistence of cycles of a continuous map of the line into itself [J]. International Journal of Bifurcation and Chaos, 1995, 5: 1263–1273.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Marotto F R. Snap-back repellers imply chaos in R n [J]. Journal of Mathematical Analysis and Applications, 1978, 63: 199–223.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Llibre J, MacKay R S. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity [J]. Ergodic Theory and Dynamical Systems, 1991, 11: 115–128.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Denvir J, MacKay R S. Consequences of contractible geodesics [J]. Transaction American Mathematical Society, 1998, 350: 4553–4568.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Boyland P L. Topological methods in surface dynamics [J]. Topology and Applications, 1994, 58: 223–298.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Li X, Chen G, Ko K T. Transition to chaos in complex dynamical networks [J]. Physica A, 2004, 338(3–4): 367–378.CrossRefMathSciNetGoogle Scholar

Copyright information

© Shanghai University 2007

Authors and Affiliations

  • Wu Rui-xin  (武瑞馨)
    • 1
  • Zhang Hai-feng  (张海峰)
    • 1
  • Fu Xin-chu  (傅新楚)
    • 2
  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiP. R. China
  2. 2.Department of Mathematics, College of SciencesShanghai UniversityShanghaiP. R. China

Personalised recommendations