SCIENTIA SINICA Physica, Mechanica & Astronomica, Volume 43, Issue 4: 467-477(2013) https://doi.org/10.1360/132012-733

Stability and Hopf bifurcation control of a fractional-order small world network model

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  • AcceptedAug 24, 2012
  • PublishedApr 15, 2013
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In this paper, a fractional-order model is firstly proposed for a small world network with time-delay, where the fractional-order derivative is used to reflect the self-similarity of the network. Then by using the method of stability switches, the stability and Hopf bifurcation of the generalized small world network with time-delay are studied. Explicit conditions for describing the stability interval and emergence of Hopf bifurcation are obtained. Further, the Pyragas type delayed feedback control is used to delay the onset of Hopf bifurcation by increasing the gain and changing the fractional-order. Numerical examples show that the stability of the controlled system can be improved substantially.


[1] Erdös P, Rényi A. On random graphs. Publications Math, 1959, 6: 290-297. Google Scholar

[2] Albert R, Barabási A L. Statistical mechanics of complex network. Rev Mod Phys, 2002, 74: 47-97. Google Scholar

[3] Watts D J, Strogatz S H. Collective dynamics of small-world networks. Nature, 1998, 393: 440-442. Google Scholar

[4] Newman M E J, Watts D J. Scaling and percolation in the small-world network model. Phys Rev E, 1999, 60: 7332-7342. Google Scholar

[5] Newman M E J, Moore C, Watts D J. Mean-field solution of the small-world network model. Phys Rev Lett, 2000, 84: 3201-3204. Google Scholar

[6] Moukarzel C F. Spreading and shortest paths in the with sparse long-range connections. Phys Rev E, 2000, 60: 6263-6266. Google Scholar

[7] Yang X S. Chaos in small-world networks. Phys Rev E, 2001, 63(4): 046206. Google Scholar

[8] Li C G, Chen G R. Local stability and Hopf bifurcation in small-world delayed networks. Chaos Solitons Fractals, 2004, 20: 353-361. Google Scholar

[9] Xiao M, Ho D W C, Cao J D. Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. Nonlinear Dyn, 2009,58(1-2): 319-344. Google Scholar

[10] Zhao H Y, Xie W. Hopf bifurcation for a small-world network model with parameters delay feedback control. Nonlinear Dyn, 2010, 05: 1-13. Google Scholar

[11] Tenreiro M J, Kiryakova V, Mainardi F. Recent history of fractional calculus. Commu Nonlinear Sci Numer Simul, 2011, 3(16): 1140-1153. Google Scholar

[12] 同登科, 王瑞和, 杨河山. 管内非Newton 流体分数阶流动的精确解. 中国科学: 物理学力学天文学, 2005, 35(3): 318-326. Google Scholar

[13] Tenreiro M J A, Silva M F, Barbosa R S, et al. Some applications of fractional calculus in engineering. Math Probl Eng, 2010, 2010: 639801,. CrossRef Google Scholar

[14] Caponetto R, Dongola G, Fortuna L, et al. Fractional Order Systems: Modeling and Control Applications. Singapore: World Scientific, 2010. Google Scholar

[15] Chen H S, Hou T T, Feng Y P. Fractional model for the physical aging of polymers (in Chinese). Sci Sin-Phys Mech Astron, 2010, 40(10):1267-1274 [陈宏善, 侯婷婷, 冯养平. 聚合物物理老化的分数阶模型. 中国科学: 物理学力学天文学, 2010, 40(10): 1267-1274]. Google Scholar

[16] Chen N, Chen N, Chen Y D. On fractional control method for four-wheel-steering vehicle. Sci China Ser E-Tech Sci, 2009, 52(3): 603-609 [陈宁, 陈南, 陈炎东. 四轮转向车辆分数阶控制方法研究. 中国科学: 技术科学, 2010, 40(2): 139-144]. Google Scholar

[17] Wang Z H, Hu H Y. Stability of a linear oscillator with damping force of fractional-order derivative (in Chinese). Sci Sin-Phys Mech Astron,2009, 39(10): 1495-1502 [王在华, 胡海岩. 含分数阶导数阻尼的线性振动系统的稳定性. 中国科学: 物理学力学天文学, 2009,39(10): 1495-1502]. Google Scholar

[18] Lundstrom B N, Higgs M H, Spain W J, et al. Fractional differentiation by neocortical pyramidal neurons. Nat Neurosci, 2008, 11: 1335-1342. Google Scholar

[19] Thomas J A. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol Cybern, 1994, 72: 69-79. Google Scholar

[20] Nakagawa M, Sorimachi K. Basic characteristics of a fractance device. IEICE Trans Fundam Electron Commun Comput Sci, 1992, E75-A(12):1814-1819. Google Scholar

[21] Wang X F, Chen G. Synchronization in small-world dynamical networks. Int J Bifur Chaos, 2002, 12: 187-192. Google Scholar

[22] Wang X F, Chen G. Complex networks: Small-world, scale-free, and beyond. IEEE Circ Syst Mag, 2003, 3(1): 6-20. Google Scholar

[23] Hong H, Choi M Y, Kim B J. Synchronization on small-world networks. Phys Rev E, 2002, 65(2): 026139. Google Scholar

[24] Song C M, Havlin S, Makse H A. Self-similarity of complex networks. Nature, 2005, 433: 392-395. Google Scholar

[25] Moon F C, Chaotic and Fractal Dynamics. New York: Wiley, 1992. Google Scholar

[26] 徐明瑜, 谭文长. 中间过程、临界现象—- 分数阶算子理论、方法、进展及其在现代力学中的应用. 中国科学: 物理学力学天文学, 2006, 36(3): 225-238. Google Scholar

[27] Deng W H, Li C P, L¨u J H. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn, 2007, 48: 409-416. Google Scholar

[28] Ostalczyk P. Nyquist characteristics of a fractional order integrator. J Fractional Calculus, 2001, 19: 67-78. Google Scholar

[29] Shi M, Wang Z H. An effective analytical criterion for stability testing of fractional-delay systems. Automatica, 2011, 47(9): 2001-2005. Google Scholar

[30] Wang Z H, Du M L, Shi M. Stability test of fractional-delay systems via integration. Sci Sin-Phys Mech Astron, 2011, 54(10): 1839-1846. Google Scholar

[31] Farshad M B, Masoud K G. On the essential instabilities caused by fractional-order transfer functions. Math Probl Eng, 2008, 13: 419046. Google Scholar

[32] Ding D, Zhu J, Luo X. Delay induced Hopf bifurcation in a dual model of internet congestion control algorithm. Nonlinear Anal-RealWorld Appl,2009, 10(5): 2873-2883. Google Scholar

[33] Monje C A, Chen Y Q, Vinagre B M, et al. Fractional Order Systems and Controls: Fundamentals and Applications. Berlin: Springer, 2010. Google Scholar

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