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

Min XIAO 1,2,*,
• AcceptedMay 30, 2019
• PublishedDec 25, 2019
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### 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).

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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

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