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Hybrid control of delay induced hopf bifurcation of dynamical small-world network

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Abstract

In this paper, we focus on the Hopf bifurcation control of a small-world network model with time-delay. With emphasis on the relationship between the Hopf bifurcation and the time-delay, we investigate the effect of time-delay by choosing it as the bifurcation parameter. By using tools from control and bifurcation theory, it is proved that there exists a critical value of time-delay for the stability of the model. When the time-delay passes through the critical value, the model loses its stability and a Hopf bifurcation occurs. To enhance the stability of the model, we propose an improved hybrid control strategy in which state feedback and parameter perturbation are used. Through linear stability analysis, we show that by adjusting the control parameter properly, the onset of Hopf bifurcation of the controlled model can be delayed or eliminated without changing the equilibrium point of the model. Finally, numerical simulations are given to verify the theoretical analysis.

Key words

Hopf bifurcation hybrid control small-world network bifurcation control 

CLC number

O 193 

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References

  1. [1]
    BARABáSI A L, ALBERT R, JEONG H. Mean field theory for scale free random graph [J]. Physica A: Statistical Mechanics and Its Applications, 1999, 272(1): 173–187.CrossRefGoogle Scholar
  2. [2]
    WATTS D J, STROGAZ S H. Collective dynamics of small-world network [J]. Nature, 1998, 393(6684): 440–442.CrossRefGoogle Scholar
  3. [3]
    BAO Z J, JIANG Q Y, YAN W J, et al. Stability of the spreading in small-world network with predictive controller [J]. Physics Letters A, 2010, 374(13): 1560–1564.CrossRefMATHGoogle Scholar
  4. [4]
    MENG M, LI S, MA H R. The transition of epidemic spreading in small world [J]. Journal of Shanghai Jiao Tong University, 2006, 40(5): 869–972 (in Chinese).Google Scholar
  5. [5]
    MOUKARZEL C F. Spreading and shortest paths in systems with sparse long-range connections [J]. Physical Review E, 1999, 60(6): 6263–6266.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    YANG X S. Fractals in small-world network with timedelay [J]. Chaos, Solitons and Fractals, 2002, 13(2): 215–219.CrossRefGoogle Scholar
  7. [7]
    LI X, CHEN G R, LI C G. Stability and bifurcation of disease spreading in complex networks [J]. International Journal of Systems Science, 2004, 35(9): 527–536.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    LI X, WANG X F. Controlling the spreading in smallworld evolving networks: Stability, oscillation, and topology [J]. IEEE Transactions on Automatic Control, 2006, 51(3): 534–540.MathSciNetCrossRefGoogle Scholar
  9. [9]
    XIAO M, HO D W C, CAO J D. Time-delayed feedback control of dynamical small-world network at Hopf bifurcation [J]. Nonlinear Dynamics, 2009, 58(1/2): 319–344.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    XU C, ZHOU Y L, WANG Y. Control of Hopf bifurcation in a fluid-flow model in wireless networks [J]. Journal of Shanghai Jiao Tong University, 2014, 48(10): 1479–1484 (in Chinese).MathSciNetGoogle Scholar
  11. [11]
    CHENG Z S, CAO J D. Hybrid control of Hopf bifurcation in complex networks with delays [J]. Neurocomputing, 2014, 131: 164–170.CrossRefGoogle Scholar
  12. [12]
    DING D W, ZHU J, LUO X S, et al. Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm [J]. Nonlinear Analysis: Real World Applications, 2009, 10(5): 2873–2883.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Dawei Ding (丁大为)
    • 1
    • 2
  • Xiaoyun Zhang (张肖芸)
    • 2
  • Nian Wang (王 年)
    • 1
    • 2
  • Dong Liang (梁 栋)
    • 1
    • 2
  1. 1.Key Laboratory of Intelligent Computing and Signal Processing of Ministry of EducationHefeiChina
  2. 2.School of Electronics and Information EngineeringAnhui UniversityHefeiChina

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