Nonlinear Dynamics

, Volume 56, Issue 1–2, pp 127–144 | Cite as

Effects of small world connection on the dynamics of a delayed ring network

  • X. Xu
  • Z. H. Wang
Original Paper


This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient “switch” to control the dynamics of a system.


Ring network Small world Global stability Bifurcation Chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pandit, S.A., Amritkar, R.E.: Characterization and control of small-world networks. Phys. Rev. E 60, 1119–1122 (1999) CrossRefGoogle Scholar
  2. 2.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Newman, M.E.J., Morre, C., Watts, D.J.: Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204 (2000) CrossRefGoogle Scholar
  4. 4.
    Wang, X.F., Chen, G.: Synchronization in small-world dynamical networks. Int. J. Bifur. Chaos 12, 187–192 (2002) CrossRefGoogle Scholar
  5. 5.
    Kulkarni, R.V., Almaas, E., Stroud, D.: Exact results and scaling properties of small-world networks. Phys. Rev. E 61, 4268–4271 (2000) CrossRefGoogle Scholar
  6. 6.
    Belykh, I.V., Belykh, V.N., Hasler, M.: Blinking model and synchronization in small-world networks with a time-varying coupling. Physica D 195, 188–206 (2004) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Zekri, N., Porterie, B., Clerc, J.P., Loraud, J.C.: Propagation in a two-dimensional weighted local small-world network. Phys. Rev. E 71, 046121–046125 (2005) CrossRefGoogle Scholar
  8. 8.
    Xu, X., Hu, H.Y., Wang, H.L.: Dynamics of a two dimensional delayed small-world network under delayed feedback control. Int. J. Bifur. Chaos 16, 3257–3273 (2006) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Yang, X.S.: Chaos in small-world networks. Phys. Rev. E 63, 046206 (2001) CrossRefGoogle Scholar
  10. 10.
    Yang, X.H.: Fractals in small-world networks with time-delay. Chaos Solitons Fractals 13, 215–219 (2002) MATHCrossRefGoogle Scholar
  11. 11.
    Li, C., Chen, G.: Local stability and Hopf bifurcation in small-world delayed networks. Chaos Solitons Fractals 20, 353–361 (2004) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998) CrossRefGoogle Scholar
  13. 13.
    Latora, V., Marchiori, M.: Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001) CrossRefGoogle Scholar
  14. 14.
    Marchiori, M., Latora, V.: Harmony in the small-world. Physica A 285, 539 (2000) MATHCrossRefGoogle Scholar
  15. 15.
    Barabasi, A.L., Jeong, H., Ravasz, R., Neda, Z., Viscek, T., Schubert, A.: Evolution of the social network of scientific collaborations. Physica A 311, 590–614 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Plois, G.A.: Stability is woven by complex webs. Nature 395, 744–745 (1998) CrossRefGoogle Scholar
  17. 17.
    McCann, K., Hastings, A., Huxel, G.R.: Weak tropic interactions and the balance of nature. Nature 395, 794–798 (1998) CrossRefGoogle Scholar
  18. 18.
    Berlow, E.L.: Strong effects of weak interactions in ecological communities. Nature 398, 330–334 (1999) CrossRefGoogle Scholar
  19. 19.
    Sporns, O., Tononi, G., Edelman, G.M.: Connectivity and complexity: the relationship between neuroanatomy and brain dynamics. Neural Netw. 13, 909 (2002) CrossRefGoogle Scholar
  20. 20.
    Latora, V., Marchiori, M.: Economic small-world behavior in weighted networks. Eur. Phys. J. B 32, 249–263 (2003) CrossRefGoogle Scholar
  21. 21.
    Campbell, S.A.: Stability and bifurcation of a simple neural network with multiple time delays. Fields Inst. Commun. 21, 65–78 (1999) Google Scholar
  22. 22.
    Campbell, S.A., Yuan, Y., Bungay, S.D.: Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18, 2827–2846 (2005) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Guo, S., Huang, L.: Hopf bifurcation periodic orbits in a ring of neurons with delays. Physica D 183, 19–44 (2003) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A: Math. Theory 41, 035102 (2008) CrossRefGoogle Scholar
  25. 25.
    Gopalsamy, K., He, X.Z.: Delay-independent stability in bidirectional associative memory networks. IEEE Trans. Neural Netw. 5, 998–1002 (1994) CrossRefGoogle Scholar
  26. 26.
    Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992) MATHGoogle Scholar
  27. 27.
    Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998) MATHCrossRefGoogle Scholar
  28. 28.
    Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Feedback Time Delays. Springer, Heidelberg (2002) Google Scholar
  29. 29.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
  30. 30.
    Ermentrout, B.: XPPAUT 3.0—The Differential Equations Tool. University of Pittsburgh, Pittsburgh (1997) Google Scholar
  31. 31.
    Balanov, A.G., Janson, N.B., Schöll, E.: Delayed feedback control of chaos: bifurcation analysis. Phys. Rev. E 71, 9 (2005) 016222-1 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.College of MathematicsJilin UniversityChangchunChina
  2. 2.Department of Mechanics at School of ScienceBeijing Institute of TechnologyBeijingChina
  3. 3.The Singapore-MIT Alliance (SMA)SingaporeSingapore
  4. 4.Institute of Vibration Engineering ResearchNanjing University of Aeronautics and AstronauticsNanjingChina

Personalised recommendations