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SCIENCE CHINA Information Sciences, Volume 63 , Issue 3 : 132201(2020) https://doi.org/10.1007/S11432-019-9946-6

Global Mittag-Leffler stability for fractional-order coupled systems on network without strong connectedness

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  • ReceivedJan 26, 2019
  • AcceptedMay 30, 2019
  • PublishedFeb 10, 2020

Abstract

This study investigates the global Mittag-Leffler stability (MLS) problem of the equilibrium point for a new fractional-order coupled system (FOCS) on a network without strong connectedness. In particular, an integer-order coupled system is extended into the FOCS on a complex network without strong connectedness. Based on the theory of asymptotically autonomous systems and graph theory, sufficient conditions are derived to ensure the existence, uniqueness, and global MLS of the solutions of this FOCS on a network. Finally, a numerical example is provided to demonstrate the validity and potential of the proposed method for studying the MLS of FOCSs.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61873071) and Shandong Provincial Natural Science Foundation (Grant No. ZR2019MF027).


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  • Figure 3

    (Color online) Response curves of (17) with the initial values $\textbf{\textit{Z}}_{1},\textbf{\textit{Z}}_{2}$, and $\textbf{\textit{Z}}_{3}$. (a) and (b) show the curves of states and control inputs of system (17).

  • Table 1   Parameters used in the coupled system (16)
    Parameter ValueParameterValueParameterValueParameter ValueParameter ValueParameter Value
    $\alpha_{1}$0.17 $\beta_{1,1}$0 $\beta_{2,1}$0.1$\beta_{3,1}$0.4$\beta_{4,1}$0$\beta_{5,1}$0
    $\alpha_{2}$0.30$\beta_{1,2}$0.1 $\beta_{2,2}$0 $\beta_{3,2}$0$\beta_{4,2}$0$\beta_{5,2}$0
    $\alpha_{3}$0.46$\beta_{1,3}$0 $\beta_{2,3}$0 $\beta_{3,3}$0$\beta_{4,3}$0.2$\beta_{5,3}$0.4
    $\alpha_{4}$0.59$\beta_{1,4}$0$\beta_{2,4}$0$\beta_{3,4}$0.2$\beta_{4,4}$0$\beta_{5,4}$0
    $\alpha_{5}$0.78$\beta_{1,5}$0$\beta_{2,5}$0$\beta_{3,5}$0$\beta_{4,5}$0$\beta_{5,5}$0
    $\alpha_{6}$0.91$\beta_{1,6}$0$\beta_{2,6}$0$\beta_{3,6}$0$\beta_{4,6}$0$\beta_{5,6}$0.4
    $\alpha_{7}$0.98 $\beta_{1,7}$0$\beta_{2,7}$0$\beta_{3,7}$0$\beta_{4,7}$0$\beta_{5,7}$0.4
    $\alpha_{8}$1.11$\beta_{1,8}$0$\beta_{2,8}$0$\beta_{3,8}$0$\beta_{4,8}$0$\beta_{5,8}$0
    $\alpha_{9}$1.21$\beta_{1,9}$0$\beta_{2,9}$ 0$\beta_{3,9}$0$\beta_{4,9}$0.4$\beta_{5,9}$0
    $\alpha_{10}$1.34$\beta_{1,10}$0$\beta_{2,10}$0$\beta_{3,10}$0$\beta_{4,10}$0$\beta_{5,10}$0
    $\beta_{6,1}$0$\beta_{7,1}$0$\beta_{8,1}$0$\beta_{9,1}$0$\beta_{10,1}$0
    $\beta_{6,2}$ 0$\beta_{7,2}$0.4$\beta_{8,2}$0$\beta_{9,2}$0 $\beta_{10,2}$0
    $\beta_{6,3}$0$\beta_{7,3}$0$\beta_{8,3}$0$\beta_{9,3}$0$\beta_{10,3}$0
    $\beta_{6,4}$0.4$\beta_{7,4}$0$\beta_{8,4}$0$\beta_{9,4}$0$\beta_{10,4}$0
    $\beta_{6,5}$0.4$\beta_{7,5}$0 $\beta_{8,5}$0$\beta_{9,5}$0$\beta_{10,5}$0
    $\beta_{6,6}$0$\beta_{7,6}$0$\beta_{8,6}$0$\beta_{9,6}$0$\beta_{10,6}$0
    $\beta_{6,7}$0$\beta_{7,7}$0$\beta_{8,7}$0.2$\beta_{9,7}$0 $\beta_{10,7}$0
    $\beta_{6,8}$0.4$\beta_{7,8}$0.2$\beta_{8,8}$0$\beta_{9,8}$0$\beta_{10,8}$0
    $\beta_{6,9}$0$\beta_{7,9}$0 $\beta_{8,9}$0$\beta_{9,9}$0$\beta_{10,9}$0.1
    $\beta_{6,10}$0$\beta_{7,10}$0$\beta_{8,10}$0.4$\beta_{9,10}$0.1$\beta_{10,10}$ 0
  • Table 2   Equilibrium values of $\textbf{\textit{Z}}^{\ast}$ of (17) for different values of $m_{k}$
    Parameter $\textit{m}_{k}=1$ $\textit{m}_{k}=2$ $\textit{m}_{k}=3$ $\textit{m}_{k}=4$ Parameter $\textit{m}_{k}=1$ $\textit{m}_{k}=2$ $\textit{m}_{k}=3$ $\textit{m}_{k}=4$
    $\textit{x}^{\ast}_{1}$ 0.1755 0.0982 0.0680 0.0520 $\textit{u}^{\ast}_{1}$ 0.1595 0.1785 0.1854 0.1890
    $\textit{x}^{\ast}_{2}$ 0.3199 0.1959 0.1401 0.1088 $\textit{u}^{\ast}_{2}$ 0.2666 0.3265 0.3503 0.3627
    $\textit{x}^{\ast}_{3}$ 0.4110 0.2765 0.2066 0.1644 $\textit{u}^{\ast}_{3}$ 0.3161 0.4254 0.4769 0.5058
    $\textit{x}^{\ast}_{4}$ 0.4691 0.3371 0.2618 0.2133 $\textit{u}^{\ast}_{4}$ 0.3351 0.4816 0.5609 0.6095
    $\textit{x}^{\ast}_{5}$ 0.5045 0.3870 0.3124 0.2611 $\textit{u}^{\ast}_{5}$ 0.3363 0.5159 0.6248 0.6963
    $\textit{x}^{\ast}_{6}$ 0.5560 0.4424 0.3660 0.3112 $\textit{u}^{\ast}_{6}$ 0.3475 0.5530 0.6862 0.7779
    $\textit{x}^{\ast}_{7}$ 0.6302 0.5115 0.4293 0.3689 $\textit{u}^{\ast}_{7}$ 0.3707 0.6018 0.7575 0.8680
    $\textit{x}^{\ast}_{8}$ 0.6496 0.5407 0.4628 0.4040 $\textit{u}^{\ast}_{8}$ 0.3609 0.6007 0.7713 0.8978
    $\textit{x}^{\ast}_{9}$ 0.6949 0.5882 0.5101 0.4500 $\textit{u}^{\ast}_{9}$ 0.3657 0.6192 0.8054 0.9473
    $\textit{x}^{\ast}_{10}$ 0.7136 0.6153 0.5413 0.4830 $\textit{u}^{\ast}_{10}$ 0.3568 0.6153 0.8119 0.9660

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