Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network

doi:10.1007/s12204-017-1823-7

Journal of shanghai Jiaotong University (Science) ›› 2017, Vol. 22 ›› Issue (2): 206-215.doi: 10.1007/s12204-017-1823-7

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Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network

DING Daweia,b* (丁大为), ZHANG Xiaoyunb (张肖芸), WANG Niana,b* (王年), LIANG Donga,b (梁栋)

(a. Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education; b. School of Electronics and Information Engineering, Anhui University, Hefei 230039, China)
(a. Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education; b. School of Electronics and Information Engineering, Anhui University, Hefei 230039, China)

Online:2017-03-31 Published:2017-04-04
Contact: DING Dawei (丁大为), WANG Nian(王年) E-mail:dwding@ahu.edu.cn, wn xlb@ahu.edu.cn

Abstract

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

摘要： 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.

关键词: Hopf bifurcation| hybrid control| small-world network| bifurcation control

CLC Number:

O 193

DING Daweia,b* (丁大为), ZHANG Xiaoyunb (张肖芸), WANG Niana,b* (王年), LIANG Donga,b (梁栋). Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network[J]. Journal of shanghai Jiaotong University (Science), 2017, 22(2): 206-215.

References 12

[1]	BARAB′ASI A L, ALBERT R, JEONG H. Mean fieldtheory for scale free random graph [J]. Physica A: StatisticalMechanics and Its Applications, 1999, 272(1):173-187.
[2]	WATTS D J, STROGAZ S H. Collective dynamics ofsmall-world network [J]. Nature, 1998, 393(6684): 440-442.
[3]	BAO Z J, JIANG Q Y, YAN W J, et al. Stability ofthe spreading in small-world network with predictivecontroller [J]. Physics Letters A, 2010, 374(13): 1560-1564.
[4]	MENG M, LI S, MA H R. The transition of epidemicspreading in small world [J]. Journal of Shanghai JiaoTong University, 2006, 40(5): 869-972 (in Chinese).
[5]	MOUKARZEL C F. Spreading and shortest paths insystems with sparse long-range connections [J]. PhysicalReview E, 1999, 60(6): 6263-6266.
[6]	YANG X S. Fractals in small-world network with timedelay[J]. Chaos, Solitons and Fractals, 2002, 13(2):215-219.
[7]	LI X, CHEN G R, LI C G. Stability and bifurcationof disease spreading in complex networks [J]. InternationalJournal of Systems Science, 2004, 35(9): 527-536.
[8]	LI X, WANG X F. Controlling the spreading in smallworldevolving networks: Stability, oscillation, andtopology [J]. IEEE Transactions on Automatic Control,2006, 51(3): 534-540.
[9]	XIAO M, HO D W C, CAO J D. Time-delayed feedbackcontrol of dynamical small-world network at Hopfbifurcation [J]. Nonlinear Dynamics, 2009, 58(1/2):319-344.
[10]	XU C, ZHOU Y L, WANG Y. Control of Hopf bifurcationin a fluid-flow model in wireless networks[J]. Journal of Shanghai Jiao Tong University, 2014,48(10): 1479-1484 (in Chinese).
[11]	CHENG Z S, CAO J D. Hybrid control of Hopf bifurcationin complex networks with delays [J]. Neurocomputing,2014, 131: 164-170.
[12]	DING D W, ZHU J, LUO X S, et al. Delay inducedHopf bifurcation in a dual model of Internet congestioncontrol algorithm [J]. Nonlinear Analysis: Real WorldApplications, 2009, 10(5): 2873-2883.

Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network

Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network

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