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Bifurcation control in the delayed fractional competitive web-site model with incommensurate-order

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Abstract

The delayed competitive web-site system with incommensurate fractional orders, based on the Lotka–Volterra competition model, is firstly proposed in this paper. It is demonstrated that there is a stability switch for time delay, Hopf bifurcation occurs when time delay crosses through a critical value and each order has important influence on the creation of bifurcation. Furthermore, a nonlinear delayed feedback control is successfully designed to postpone the onset of Hopf bifurcation, extend the stability domain, and then the system possesses the stability in a larger parameter range. Finally, numerical simulations are included to illustrate the efficiency of the obtained theoretical results.

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Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61573096, 61272530 and 61573194, the National Science Foundational of Jiangsu Province of China under Grant no. BK2012741, the “333 Engineering” Foundation of Jiangsu Province of China under Grant no. BRA2015286.

Author information

Authors and Affiliations

  1. School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, 211171, China

    Lingzhi Zhao

  2. Research Center for Complex Systems and Network Sciences, and School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Jinde Cao, Ahmed Alsaedi & Bashir Ahmad

  4. School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, 441053, China

    Chengdai Huang

  5. College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

    Min Xiao

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  1. Lingzhi Zhao
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  2. Jinde Cao
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  3. Chengdai Huang
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  4. Min Xiao
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Correspondence to Lingzhi Zhao.

Appendix

Appendix

1.1 Appendix A

$$\begin{aligned} \mathfrak {M}_1&=-\alpha x^*\omega _0^{q_2+q_3+1}\sin \frac{(q_2+q_3)\pi }{2}-\alpha ^2{(x^*)}^2 \left(\omega _0^{q_2+1}\sin \frac{q_2\pi }{2}+\omega _0^{q_3+1}\sin \frac{q_3\pi }{2}\right),\\ \mathfrak {M}_2&=\omega _0\alpha ^3{(x^*)}^3-\omega _0\alpha \gamma ^2{(x^*)}^3+\alpha x^*\omega _0^{q_2+q_3+1}\cos \frac{(q_2+q_3)\pi }{2}\\&\quad+\alpha ^2{(x^*)}^2\left(\omega _0^{q_2+1}\cos \frac{q_2\pi }{2}+\omega _0^{q_3+1} \cos \frac{q_3\pi }{2}\right),\\ \mathfrak {N}_1&=\alpha x^*(q_2+q_3)\omega _0^{q_2+q_3-1}\cos \frac{(q_2+q_3-1)\pi }{2}\\&\quad+\alpha ^2{(x^*)}^2\left[q_2\omega _0^{q_2-1}\cos \frac{(q_2-1)\pi }{2} +q_3\omega _0^{q_3-1}\cos \frac{(q_3-1)\pi }{2}\right]\\&\quad-\tau _0\left[\alpha ^3{(x^*)}^3-\alpha \gamma ^2{(x^*)}^3+\alpha x^*\omega _0^{q_2+q_3}\cos \frac{(q_2+q_3)\pi }{2}\right.\\ &\quad\left.+\alpha ^2{(x^*)}^2\left(\omega _0^{q_2}\cos \frac{q_2\pi }{2}+\omega _0^{q_3}\cos \frac{q_3\pi }{2}\right)\right] +\left[(q_1+q_2+q_3) \right.\\&\quad\times\omega _0^{q_1+q_2+q_3-1}\cos \frac{(q_1+q_2+q_3-1)\pi }{2}+ \alpha x^*(q_1+q_2)\omega _0^{q_1+q_2-1}\\&\quad\times\cos \frac{(q_1+q_2-1)\pi }{2} +\alpha x^*(q_1+q_3)\omega _0^{q_1+q_3-1} \\&\quad\times\cos \frac{(q_1+q_3-1)\pi }{2}+(\alpha ^2{(x^*)}^2 -\gamma ^2{(x^*)}^2)~q_1\omega _0^{q_1-1} \cos \frac{(q_1-1)\pi }{2}\\&\quad-\gamma ^2{(x^*)}^2 q_2\omega _0^{q_2-1}\cos \frac{(q_2-1)\pi }{2} -\gamma ^2{(x^*)}^2q_3\omega _0^{q_3-1}\\&\quad\left.\times\cos \frac{(q_3-1)\pi }{2})\right]\cos \omega _0\tau _0 -\left[(q_1+q_2+q_3)\omega _0^{q_1+q_2+q_3-1}\sin \frac{(q_1+q_2+q_3-1)\pi }{2}\right.\\&\quad+\alpha x^*(q_1+q_2)\omega _0^{q_1+q_2-1}\sin \frac{(q_1+q_2-1)\pi }{2}+\alpha x^*(q_1+q_3)\omega _0^{q_1+q_3-1}\sin \frac{(q_1+q_3-1)\pi }{2}\\&\quad+(\alpha ^2{(x^*)}^2 -\gamma ^2{(x^*)}^2)~q_1\omega _0^{q_1-1} \sin \frac{(q_1-1)\pi }{2}\\&\quad\left.-\gamma ^2{(x^*)}^2 q_2\omega _0^{q_2-1}\sin \frac{(q_2-1)\pi }{2}-\gamma ^2{(x^*)}^2q_3\omega _0^{q_3-1} \sin \frac{(q_3-1)\pi }{2})\right]\sin \omega _0\tau _0, \end{aligned}$$
$$\begin{aligned} \mathfrak {N}_2&=\alpha x^*(q_2+q_3)\omega _0^{q_2+q_3-1}\sin \frac{(q_2+q_3-1)\pi }{2} +\alpha ^2{(x^*)}^2\left[q_2\omega _0^{q_2-1}\right.\\&\quad\left.\times\sin \frac{(q_2-1)\pi }{2} +q_3\omega _0^{q_3-1}\sin \frac{(q_3-1)\pi }{2}\right]-\tau _0\left[\alpha ^3{(x^*)}^3-\alpha \gamma ^2{(x^*)}^3\right.\\&\quad\left.+\alpha x^*\omega _0^{q_2+q_3}\sin \frac{(q_2+q_3)\pi }{2} +\alpha ^2{(x^*)}^2(\omega _0^{q_2}\sin \frac{q_2\pi }{2}+\omega _0^{q_3} \sin \frac{q_3\pi }{2})\right] +\left[(q_1+q_2+q_3) \right.\\&\quad\times\omega _0^{q_1+q_2+q_3-1}\cos \frac{(q_1+q_2+q_3-1)\pi }{2}+ \alpha x^*(q_1+q_2)\omega _0^{q_1+q_2-1}\\&\quad\times\cos \frac{(q_1+q_2-1)\pi }{2} +\alpha x^*(q_1+q_3)\omega _0^{q_1+q_3-1} \\&\quad\times\cos \frac{(q_1+q_3-1)\pi }{2}+(\alpha ^2{(x^*)}^2-\gamma ^2{(x^*)}^2) ~q_1\omega _0^{q_1-1} \cos \frac{(q_1-1)\pi }{2}\\&\quad-\gamma ^2{(x^*)}^2 q_2\omega _0^{q_2-1}\cos \frac{(q_2-1)\pi }{2} -\gamma ^2{(x^*)}^2q_3\omega _0^{q_3-1}\\&\quad\left.\times\cos \frac{(q_3-1)\pi }{2}\right]\sin \omega _0\tau _0 +\left[(q_1+q_2+q_3)\omega _0^{q_1+q_2+q_3-1}\right.\\&\quad\times\sin \frac{(q_1+q_2+q_3-1)\pi }{2}+\alpha x^*(q_1+q_2)\omega _0^{q_1+q_2-1}\\&\quad\times\sin \frac{(q_1+q_2-1)\pi }{2}+\alpha x^*(q_1+q_3)\omega _0^{q_1+q_3-1}\sin \frac{(q_1+q_3-1)\pi }{2}\\&\quad+(\alpha ^2{(x^*)}^2 -\gamma ^2{(x^*)}^2)~q_1\omega _0^{q_1-1} \sin \frac{(q_1-1)\pi }{2}\\&\quad\left.-\gamma ^2{(x^*)}^2 q_2\omega _0^{q_2-1}\sin \frac{(q_2-1)\pi }{2}-\gamma ^2{(x^*)}^2q_3\omega _0^{q_3-1} \sin \frac{(q_3-1)\pi }{2}\right]\cos \omega _0\tau _0. \end{aligned}$$

1.2 Appendix B

$$\begin{aligned}\mathfrak {M}^*_1&=\omega _0^*(\alpha ^3{x^*}^3-\alpha \gamma ^2{x^*}^3 +k_1\alpha ^2{x^*}^2)+\alpha x^*{(\omega _0^*)}^{q_2+q_3+1}\cos \frac{(q_2+q_3)\pi }{2}\\&\quad+\alpha ^2{(x^*)}^2\left[(\omega _0^*)^{q_2+1}\cos \frac{q_2\pi }{2}+(\omega _0^*)^{q_3+1} \cos \frac{q_3\pi }{2}\right]\\&\quad+k_1\alpha x^*(\omega _0^*)^{q_3+1}\cos \frac{q_3\pi }{2},\\ \mathfrak {M}^*_2&=\alpha x^*(\omega _0^*)^{q_2+q_3+1}\sin \frac{(q_2+q_3)\pi }{2}+\alpha ^2{(x^*)}^2 \left[(\omega _0^*)^{q_2+1}\sin \frac{q_2\pi }{2}\right.\\&\quad\left.+(\omega _0^*)^{q_3+1} \sin \frac{q_3\pi }{2}+k_1\alpha x^*(\omega _0^*)^{q_3+1}\sin \frac{q_3\pi }{2}\right],\\ \mathfrak {N}^*_1&=\alpha x^*(q_2+q_3)(\omega _0^*)^{q_2+q_3-1}\cos \frac{(q_2+q_3-1)\pi }{2} +\alpha ^2{(x^*)}^2\left[q_2(\omega _0^*)^{q_2-1}\right.\\&\quad\left.\times\cos \frac{(q_2-1)\pi }{2} +q_3(\omega _0^*)^{q_3-1}\cos \frac{(q_3-1)\pi }{2}\right] \\&\quad+k_1\alpha x^*q_3(\omega _0^*)^{q_3-1} \cos \frac{(q_3-1)\pi }{2}-\tau _0^*\left[\alpha ^3{(x^*)}^3-\alpha \gamma ^2{(x^*)}^3\right.\\&\quad+\alpha x^*(\omega _0^*)^{q_2+q_3}\cos \frac{(q_2+q_3)\pi }{2} +\alpha ^2{(x^*)}^2\left((\omega _0^*)^{q_2}\cos \frac{q_2\pi }{2}\right.\\&\quad\left.\left.+(\omega _0^*)^{q_3}\cos \frac{q_3\pi }{2}\right) +k_1\alpha x^*(\omega _0^*)^{q_3}\cos \frac{q_3\pi }{2}\right] +\left[(q_1+q_2+q_3)(\omega _0^*)^{q_1+q_2+q_3-1}\right.\\&\quad\times\cos \frac{(q_1+q_2+q_3-1)\pi }{2}+ \alpha x^*(q_1+q_2)\\&\quad\times(\omega _0^*)^{q_1+q_2-1}\cos \frac{(q_1+q_2-1)\pi }{2} +\alpha x^*(q_1+q_3)(\omega _0^*)^{q_1+q_3-1} \cos \frac{(q_1+q_3-1)\pi }{2}\\&\quad+k_1(q_1+q_3)(\omega _0^*)^{q_1+q_3-1} \cos \frac{(q_1+q_3-1)\pi }{2}\\&\quad+(\alpha ^2{(x^*)}^2-\gamma ^2{(x^*)}^2+k_1\alpha x^*)~q_1(\omega _0^*)^{q_1-1} \cos \frac{(q_1-1)\pi }{2}\\&\quad\left.-\gamma ^2{x^*}^2q_2 (\omega _0^*)^{q_2-1}\cos \frac{(q_2-1)\pi }{2 }-\gamma ^2{(x^*)}^2q_3(\omega _0^*)^{q_3-1}\cos \frac{(q_3-1)\pi }{2}\right]\\&\quad\times\cos \omega _0^*\tau _0^*-\left[(q_1+q_2+q_3)(\omega _0^*)^{q_1+q_2+q_3-1} \sin \frac{(q_1+q_2+q_3-1)\pi }{2}\right.\\&\quad+\alpha x^*(q_1+q_2)s^{q_1+q_2-1}\sin \frac{(q_1+q_2-1)\pi }{2}+\alpha x^*(q_1+q_3)\\&\quad\times(\omega _0^*)^{q_1+q_3-1}\sin \frac{(q_1+q_3-1)\pi }{2} +(\alpha ^2{(x^*)}^2-\gamma ^2{(x^*)}^2)~q_1(\omega _0^*)^{q_1-1} \sin \frac{(q_1-1)\pi }{2}\\&\quad-\gamma ^2{(x^*)}^2 q_2(\omega _0^*)^{q_2-1}\sin \frac{(q_2-1)\pi }{2}-\gamma ^2{(x^*)}^2q_3\\&\quad\left.\times(\omega _0^*)^{q_3-1} \sin \frac{(q_3-1)\pi }{2})\right]\sin \omega _0^*\tau _0^*, \end{aligned}$$
$$\begin{aligned} \mathfrak {N}^*_2&=\alpha x^*(q_2+q_3)(\omega _0^*)^{q_2+q_3-1}\sin \frac{(q_2+q_3-1)\pi }{2} +\alpha ^2{(x^*)}^2\left[q_2(\omega _0^*)^{q_2-1}\sin \frac{(q_2-1)\pi }{2} +q_3(\omega _0^*)^{q_3-1}\sin \frac{(q_3-1)\pi }{2}\right]\\ &\quad+k_1\alpha x^*q_3(\omega _0^*)^{q_3-1}\sin \frac{(q_3-1)\pi }{2}-\tau _0^*\left[\alpha ^3{(x^*)}^3 -\alpha \gamma ^2{(x^*)}^3+\alpha x^*(\omega _0^*)^{q_2+q_3}\sin \frac{(q_2+q_3)\pi }{2} +\alpha ^2{x^*}^2\left((\omega _0^*)^{q_2}\sin \frac{q_2\pi }{2}\right.\right.\\ &\quad\left.\left.+(\omega _0^*)^{q_3} \sin \frac{q_3\pi }{2}\right)+k_1\alpha x^*(\omega _0^*)^{q_3}\sin \frac{q_3\pi }{2}\right] +\left[(q_1+q_2+q_3)(\omega _0^*)^{q_1+q_2+q_3-1}\cos \frac{(q_1+q_2+q_3-1)\pi }{2}+ \alpha x^*(q_1+q_2)\right.\\ &\quad\times(\omega _0^*)^{q_1+q_2-1}\cos \frac{(q_1+q_2-1)\pi }{2} +\alpha x^*(q_1+q_3)(\omega _0^*)^{q_1+q_3-1} \cos \frac{(q_1+q_3-1)\pi }{2}+k_1(q_1+q_3)(\omega _0^*)^{q_1+q_3-1} \cos \frac{(q_1+q_3-1)\pi }{2}\\&\quad \left.+(\alpha ^2{x^*}^2 -\gamma ^2{(x^*)}^2+k_1\alpha x^*) ~q_1(\omega _0^*){q_1-1}\cos \frac{(q_1-1)\pi }{2} -\gamma ^2{(x^*)}^2q_2(\omega _0^*)^{q_2-1}\cos \frac{(q_2-1)\pi }{2}- \gamma ^2{(x^*)}^2q_3(\omega _0^*)^{q_3-1}\cos \frac{(q_3-1)\pi }{2}\right]\\ &\quad\times\sin \omega _0^*\tau _0^* +\left[(q_1+q_2+q_3)(\omega _0^*)^{q_1+q_2+q_3-1}\sin \frac{(q_1+q_2+q_3-1)\pi }{2} +\alpha x^*(q_1+q_2)s^{q_1+q_2-1}\sin \frac{(q_1+q_2-1)\pi }{2}+\alpha x^*(q_1+q_3)\right.\\&\quad\times (\omega _0^*)^{q_1+q_3-1}\sin \frac{(q_1+q_3-1)\pi }{2} +(\alpha ^2{(x^*)}^2-\gamma ^2{(x^*)}^2)~q_1(\omega _0^*)^{q_1-1} \sin \frac{(q_1-1)\pi }{2}-\gamma ^2{(x^*)}^2 q_2(\omega _0^*)^{q_2-1}\sin \frac{(q_2-1)\pi }{2}-\gamma ^2{(x^*)}^2q_3\\ &\quad\left.\times(\omega _0^*)^{q_3-1}\sin \frac{(q_3-1)\pi }{2}\right]\cos \omega _0^*\tau _0^*. \end{aligned}$$

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Zhao, L., Cao, J., Huang, C. et al. Bifurcation control in the delayed fractional competitive web-site model with incommensurate-order. Int. J. Mach. Learn. & Cyber. 10, 173–186 (2019). https://doi.org/10.1007/s13042-017-0707-3

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