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



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).


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  • 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