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.
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References
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.
WATTS D J, STROGAZ S H. Collective dynamics of small-world network [J]. Nature, 1998, 393(6684): 440–442.
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.
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).
MOUKARZEL C F. Spreading and shortest paths in systems with sparse long-range connections [J]. Physical Review E, 1999, 60(6): 6263–6266.
YANG X S. Fractals in small-world network with timedelay [J]. Chaos, Solitons and Fractals, 2002, 13(2): 215–219.
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.
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.
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.
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).
CHENG Z S, CAO J D. Hybrid control of Hopf bifurcation in complex networks with delays [J]. Neurocomputing, 2014, 131: 164–170.
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.
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Foundation item: the National Natural Science Foundation of China (Nos. 61201227 and 61172127), and the Natural Science Foundation of Anhui (No. 1208085MF93)
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Ding, D., Zhang, X., Wang, N. et al. Hybrid control of delay induced hopf bifurcation of dynamical small-world network. J. Shanghai Jiaotong Univ. (Sci.) 22, 206–215 (2017). https://doi.org/10.1007/s12204-017-1823-7
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DOI: https://doi.org/10.1007/s12204-017-1823-7