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SCIENCE CHINA Information Sciences, Volume 63 , Issue 1 : 112206(2020) https://doi.org/10.1007/s11432-018-9933-6

Improving dynamics of integer-order small-world network models under fractional-order PD control

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  • ReceivedNov 7, 2018
  • AcceptedMay 30, 2019
  • PublishedDec 25, 2019

Abstract

The optimal control of dynamics is a popular topic for small-world networks. In this paper, we address the problem of improving the behavior of Hopf bifurcations in an integer-order model of small-world networks. In this study, the time delay is used as the bifurcation parameter. We add a fractional-order proportional-derivative (PD) scheme to an integer-order Newman-Watts (N-W) small-world model to better control the Hopf bifurcation of the model. The most important contribution of this paper involves obtaining the stability of the system and the variation of the conditions of the Hopf bifurcation after a fractional PD controller is added to the integer-order small-world model. The results demonstrate that the designed PD controller can be used to restrain or promote the occurrence of Hopf bifurcations by setting appropriate parameters. We also describe several simulations to verify our research results.


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61573194, 51775284, 61877033), Natural Science Foundation of Jiangsu Province of China (Grant Nos. BK20181389, BK20181387), Key Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (Grant No. 2018SJZDI142), and Postgraduate Research Practice Innovation Program of Jiangsu Province (Grant No. KYCX18_0924).


References

[1] Watts D J, Strogatz S H. Collective dynamics of `small-world' networks. Nature, 1998, 393: 440-442 CrossRef PubMed ADS Google Scholar

[2] Newman M E J, Watts D J. Renormalization group analysis of the small-world network model. Phys Lett A, 1999, 263: 341-346 CrossRef ADS Google Scholar

[3] Yang X S. Chaos in small-world networks. Phys Rev E, 2001, 63: 046206 CrossRef PubMed ADS arXiv Google Scholar

[4] Xiao M, Ho D W C, Cao J. Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. NOnlinear Dyn, 2009, 58: 319-344 CrossRef Google Scholar

[5] Xu X, Luo J W. Dynamical model and control of a small-world network with memory. NOnlinear Dyn, 2013, 73: 1659-1669 CrossRef Google Scholar

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

[7] Li N, Sun H Y, Zhang Q L. Bifurcations and chaos control in discrete small-world networks. Chin Phys B, 2012, 21: 010503 CrossRef ADS Google Scholar

[8] Feng L, Zhi-Hong G, Hua W. GENERAL: Controlling bifurcations and chaos in discrete small-world networks. Chin Phys B, 2008, 17: 2405-2411 CrossRef ADS Google Scholar

[9] Mahajan A V, Gade P M. Transition from clustered state to spatiotemporal chaos in a small-world networks. Phys Rev E, 2010, 81: 056211 CrossRef PubMed ADS Google Scholar

[10] Wu X, Zhao X, Lu J. Identifying Topologies of Complex Dynamical Networks With Stochastic Perturbations. IEEE Trans Control Netw Syst, 2016, 3: 379-389 CrossRef Google Scholar

[11] Maslennikov O V, Nekorkin V I, Kurths J. Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators. Phys Rev E, 2015, 92: 042803 CrossRef PubMed ADS Google Scholar

[12] Mei G, Wu X, Ning D. Finite-time stabilization of complex dynamical networks via optimal control. Complexity, 2016, 21: 417-425 CrossRef ADS Google Scholar

[13] Xiao M, Zheng W X, Lin J. Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks. J Franklin Institute, 2017, 354: 7643-7667 CrossRef Google Scholar

[14] Zhou J, Xu X, Yu D. Stability, Instability and Bifurcation Modes of a Delayed Small World Network with Excitatory or Inhibitory Short-Cuts. Int J Bifurcation Chaos, 2016, 26: 1650070 CrossRef ADS Google Scholar

[15] Cao J, Guerrini L, Cheng Z. Stability and Hopf bifurcation of controlled complex networks model with two delays. Appl Math Computation, 2019, 343: 21-29 CrossRef Google Scholar

[16] Cao Y. Bifurcations in an Internet congestion control system with distributed delay. Appl Math Computation, 2019, 347: 54-63 CrossRef Google Scholar

[17] Hassard B D, Kazarinoff N D, Wan Y H. Theory and Applications of Hopf bifurcation. Cambridge: Cambridge University Press, 1981. Google Scholar

[18] Han M, Sheng L, Zhang X. Bifurcation theory for finitely smooth planar autonomous differential systems. J Differ Equ, 2018, 264: 3596-3618 CrossRef ADS Google Scholar

[19] Tian H, Han M. Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems. J Differ Equ, 2017, 263: 7448-7474 CrossRef ADS Google Scholar

[20] Wu Y, Liu L, Zhang X. Bifurcation analysis for a singular differential system with two parameters via to topological degree theory. NOnlinear Anal, 2017, 2017(1): 31-50 CrossRef Google Scholar

[21] Li S, Peng X, Tang Y. Finite-time synchronization of time-delayed neural networks with unknown parameters via adaptive control. Neurocomputing, 2018, 308: 65-74 CrossRef Google Scholar

[22] Guo W, Yang J. Hopf bifurcation control of hydro-turbine governing system with sloping ceiling tailrace tunnel using nonlinear state feedback. Chaos Solitons Fractals, 2017, 104: 426-434 CrossRef ADS Google Scholar

[23] Syed Ali M, Yogambigai J. Passivity-based synchronization of stochastic switched complex dynamical networks with additive time-varying delays via impulsive control. Neurocomputing, 2018, 273: 209-221 CrossRef Google Scholar

[24] Liu R, She J, Wu M. Robust disturbance rejection for a fractional-order system based on equivalent-input-disturbance approach. Sci China Inf Sci, 2018, 61: 070222 CrossRef Google Scholar

[25] Al Hosani K, Nguyen T H, Al Sayari N. Fault-tolerant control of MMCs based on SCDSMs in HVDC systems during DC-cable short circuits. Int J Electrical Power Energy Syst, 2018, 100: 379-390 CrossRef Google Scholar

[26] Ding D, Zhang X, Cao J. Bifurcation control of complex networks model via PD controller. Neurocomputing, 2016, 175: 1-9 CrossRef Google Scholar

[27] Tang Y, Xiao M, Jiang G. Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system. NOnlinear Dyn, 2017, 90: 2185-2198 CrossRef Google Scholar

[28] Zhang W, Dong X, Liu X. Switched Fuzzy-PD Control of Contact Forces in Robotic Microbiomanipulation.. IEEE Trans Biomed Eng, 2017, 64: 1169-1177 CrossRef PubMed Google Scholar

[29] Ouyang P R, Pano V, Tang J. Position domain nonlinear PD control for contour tracking of robotic manipulator. Robotics Comput-Integrated Manufacturing, 2018, 51: 14-24 CrossRef Google Scholar

[30] ?zbay H, Bonnet C, Fioravanti A R. PID controller design for fractional-order systems with time delays. Syst Control Lett, 2012, 61: 18-23 CrossRef Google Scholar

[31] Wu J, Zhang X, Liu L. Iterative algorithm and estimation of solution for a fractional order differential equation. Bound Value Probl, 2016, 2016(1): 116 CrossRef Google Scholar

[32] Li M, Wang J R. Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations. Appl Math Computation, 2018, 324: 254-265 CrossRef Google Scholar

[33] Zhang X, Liu L, Wu Y. The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl Math Lett, 2014, 37: 26-33 CrossRef Google Scholar

[34] Bao F, Yao X, Sun Q. Smooth fractal surfaces derived from bicubic rational fractal interpolation functions. Sci China Inf Sci, 2018, 61: 099104 CrossRef Google Scholar

[35] Guan Y, Zhao Z, Lin X. On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques. Bound Value Probl, 2016, 2016(1): 141 CrossRef Google Scholar

[36] Shao J, Zheng Z, Meng F. Oscillation criteria for fractional differential equations with mixed nonlinearities. Adv Differ Equ, 2013, 2013(1): 323 CrossRef Google Scholar

[37] Podlubny I. Fractional Differential Equations. NewYork: Academic Press, 1999. Google Scholar

[38] Li C, Deng W. Remarks on fractional derivatives. Appl Math Computation, 2007, 187: 777-784 CrossRef Google Scholar

[39] Bhalekar S, Varsha D. A predictor-corrector scheme for solving nonlinear delay differential equations of fractional ord er. J Fractional Calc Appl, 2011, 1: 1--9. Google Scholar

[40] Chen Z, Zhao D, Ruan J. Delay Induced Hopf Bifurcation of Small-World Networks. Chin Ann Math Ser B, 2007, 28: 453-462 CrossRef Google Scholar

  • Figure 3

    Bifurcation diagram of $i(t)$ vs. $\tau_{00}$ with initial values $n=1$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=0.8$, and $K_{d}=-0.1$.

  • Figure 6

    Bifurcation diagram of $i(t)$ vs. $\tau_{01}$ with initial values $n=1$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=0.8$, and $K_{d}=-0.1$.

  • Figure 9

    Bifurcation diagram of $i(t)$ vs. $\tau_{01}$ with initial values $n=1$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=0.7$, and $K_{d}=-0.5$.

  • Figure 11

    (Color online) Waveform plot of controlled model (5) with initial values $n=3$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=0.8$, and $K_{d}=-0.2$. The equilibrium is unstable when $\tau=0.68>\tau_{02}$.

  • Figure 12

    Bifurcation diagram of $i(t)$ vs. $\tau_{02}$ with initial values $n=3$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=0.8$, and $K_{d}=-0.2$.

  • Figure 15

    Bifurcation diagram of $i(t)$ vs. $\tau_{02}$ with initial values $n=3$, $i(0)=0$, $\varepsilon=3$, $\mu=0.1$, $K_{p}=-0.2$, and $K_{d}=-0.2$.

  • Table 1   Effect of $n$ on value of $\tau_0$ for controlled system (5) with $K_{p}~=-1$ and $K_{d}~=-1$
    Fractional-order parameter $n$ Bifurcation point $\tau_0$
    1 0.8855
    2 0.7856
    3 0.7761
    4 0.7735
    5 0.7720
    6 0.7711
    7 0.7706
    8 0.7703

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