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

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Abstract

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.

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References

  1. Pandit, S.A., Amritkar, R.E.: Characterization and control of small-world networks. Phys. Rev. E 60, 1119–1122 (1999)

    Article  Google Scholar 

  2. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  4. Wang, X.F., Chen, G.: Synchronization in small-world dynamical networks. Int. J. Bifur. Chaos 12, 187–192 (2002)

    Article  Google Scholar 

  5. Kulkarni, R.V., Almaas, E., Stroud, D.: Exact results and scaling properties of small-world networks. Phys. Rev. E 61, 4268–4271 (2000)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yang, X.S.: Chaos in small-world networks. Phys. Rev. E 63, 046206 (2001)

    Article  Google Scholar 

  10. Yang, X.H.: Fractals in small-world networks with time-delay. Chaos Solitons Fractals 13, 215–219 (2002)

    Article  MATH  Google Scholar 

  11. Li, C., Chen, G.: Local stability and Hopf bifurcation in small-world delayed networks. Chaos Solitons Fractals 20, 353–361 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)

    Article  Google Scholar 

  13. Latora, V., Marchiori, M.: Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001)

    Article  Google Scholar 

  14. Marchiori, M., Latora, V.: Harmony in the small-world. Physica A 285, 539 (2000)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Plois, G.A.: Stability is woven by complex webs. Nature 395, 744–745 (1998)

    Article  Google Scholar 

  17. McCann, K., Hastings, A., Huxel, G.R.: Weak tropic interactions and the balance of nature. Nature 395, 794–798 (1998)

    Article  Google Scholar 

  18. Berlow, E.L.: Strong effects of weak interactions in ecological communities. Nature 398, 330–334 (1999)

    Article  Google Scholar 

  19. Sporns, O., Tononi, G., Edelman, G.M.: Connectivity and complexity: the relationship between neuroanatomy and brain dynamics. Neural Netw. 13, 909 (2002)

    Article  Google Scholar 

  20. Latora, V., Marchiori, M.: Economic small-world behavior in weighted networks. Eur. Phys. J. B 32, 249–263 (2003)

    Article  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Guo, S., Huang, L.: Hopf bifurcation periodic orbits in a ring of neurons with delays. Physica D 183, 19–44 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A: Math. Theory 41, 035102 (2008)

    Article  Google Scholar 

  25. Gopalsamy, K., He, X.Z.: Delay-independent stability in bidirectional associative memory networks. IEEE Trans. Neural Netw. 5, 998–1002 (1994)

    Article  Google Scholar 

  26. Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992)

    MATH  Google Scholar 

  27. Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)

    Article  MATH  Google Scholar 

  28. Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Feedback Time Delays. Springer, Heidelberg (2002)

    Google Scholar 

  29. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  30. Ermentrout, B.: XPPAUT 3.0—The Differential Equations Tool. University of Pittsburgh, Pittsburgh (1997)

    Google Scholar 

  31. Balanov, A.G., Janson, N.B., Schöll, E.: Delayed feedback control of chaos: bifurcation analysis. Phys. Rev. E 71, 9 (2005) 016222-1

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics, Jilin University, 2699 Qianjin Street, 130012, Changchun, China

    X. Xu

  2. Department of Mechanics at School of Science, Beijing Institute of Technology, No. 5 Zhongguancun Nandajie, 100081, Beijing, China

    X. Xu

  3. The Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, 117576, Singapore, Singapore

    X. Xu

  4. Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Yudao Street 29, 210016, Nanjing, China

    Z. H. Wang

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  1. X. Xu
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  2. Z. H. Wang
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Correspondence to X. Xu.

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Xu, X., Wang, Z.H. Effects of small world connection on the dynamics of a delayed ring network. Nonlinear Dyn 56, 127–144 (2009). https://doi.org/10.1007/s11071-008-9384-9

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  • DOI: https://doi.org/10.1007/s11071-008-9384-9

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