Bifurcation control for a fractional-order competition model of Internet with delays

Abstract

In today’s society, the Internet has become an important tool of our life due to its potential applications in various areas such as economics, industry, agriculture, medical and health care, and information processing. To understand and grasp the law of the Internet, many competitive web site models of the Internet and some phenomena related to World Wide Web have been investigated systematically. However, many scholars only study the integer-order competitive web site models of the Internet. Up to now, there are few papers that focus on the dynamics of fractional-order competitive web site models of Internet, which possess memory property. In this paper, we are concerned with the stability and the existence of Hopf bifurcation of a fractional-order competitive web site model of Internet. By choosing the time delay as parameter and applying the Routh–Hurwitz criteria, we will establish a new sufficient condition guaranteeing the stability and the existence of Hopf bifurcation for fractional-order competitive web site model of Internet. The research reveals that fractional order and the delay play a key role in describing the stability and Hopf bifurcation of the considered system. Computer simulations are implemented to support the analytic results. Finally, a simple conclusion is presented. The theoretical findings of this article have a great significance in handling the competition dynamics among different web sites.

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Correspondence to Changjin Xu.

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This work is supported by National Natural Science Foundation of China (No. 61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651) and Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020).

Appendices

Appendix A

$$\begin{aligned} B_0= & {} u_1^*u_2^*a_1a_2a_3,\\ B_1(s)= & {} u_1^*u_2^*a_1a_2s^{p_3}+u_2^*u_3^*a_2a_3s^{p_1} \\&+\,u_1^*u_3^*a_1a_3s^{p_2},\\ B_2(s)= & {} u_2^*a_2s^{p_1+p_3}+u_1^*a_1s^{p_2+p_3}+u_3^*a_3s^{p_1+p_2}\\&-\,u_1^*u_2^*u_3^*(c_{13}c_{31}a_2+c_{12}c_{21}a_3+c_{23}c_{32}a_1),\\ B_3(s)= & {} s^{p_1+p_2+p_3}+u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31}+c_{13}c_{21}c_{32})\\&-\,u_1^*u_3^*c_{13}c_{31}s^{p_2}-u_1^*u_2^*c_{12}c_{21}s^{p_3} \\&-\,u_2^*u_3^*c_{23}c_{32}s^{p_1}. \end{aligned}$$

Appendix B

$$\begin{aligned} B_{1R}(\theta )= & {} u_1^*u_2^*a_1a_2\theta ^{p_3}\cos \frac{p_3\pi }{2}\\&+u_2^*u_3^*a_2a_3\theta ^{p_1}\cos \frac{p_1\pi }{2} \\&+\,u_1^*u_3^*a_1a_3\theta ^{p_2}\cos \frac{p_2\pi }{2},\\ B_{1I}(\theta )= & {} u_1^*u_2^*a_1a_2\theta ^{p_3}\sin \frac{p_3\pi }{2} \\&+\,u_2^*u_3^*a_2a_3\theta ^{p_1}\sin \frac{p_1\pi }{2} \\&+\,u_1^*u_3^*a_1a_3\theta ^{p_2}\sin \frac{p_2\pi }{2},\\ B_{2R}(\theta )= & {} u_2^*a_2\theta ^{p_1+p_3}\cos \frac{(p_1+p_3)\pi }{2} \\&+\,u_1^*a_1\theta ^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2} \\&+\,u_3^*a_3\theta ^{p_1+p_2}\cos \frac{(p_1+p_2)\pi }{2}\\&-\,u_1^*u_2^*u_3^*(c_{13}c_{31}a_2 \\&+\,c_{12}c_{21}a_3+c_{23}c_{32}a_1),\\ B_{2I}(\theta )= & {} u_2^*a_2\theta ^{p_1+p_3}\sin \frac{(p_1+p_3)\pi }{2} \\&+\,u_1^*a_1\theta ^{p_2+p_3}\sin \frac{(p_2+p_3)\pi }{2} \\&+\,u_3^*a_3\theta ^{p_1+p_2}\sin \frac{(p_1+p_2)\pi }{2},\\ B_{3R}(\theta )= & {} \theta ^{p_1+p_2+p_3}\cos \frac{(p_1+p_2+p_3)\pi }{2} \\&+\,u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31}+c_{13}c_{21}c_{32})\\&-\,u_1^*u_3^*c_{13}c_{31}\theta ^{p_2}\cos \frac{p_2\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}\theta ^{p_3}\cos \frac{p_3\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}\theta ^{p_1}\cos \frac{p_1\pi }{2},\\ B_{3I}(\theta )= & {} \theta ^{p_1+p_2+p_3}\cos \frac{(p_1+p_2+p_3)\pi }{2} \\&-\,u_1^*u_3^*c_{13}c_{31}\theta ^{p_2}\sin \frac{p_2\pi }{2}\\&-\,u_1^*u_2^*c_{12}c_{21}\theta ^{p_3}\sin \frac{p_3\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}\theta ^{p_1}\sin \frac{p_1\pi }{2}. \end{aligned}$$

Appendix C

$$\begin{aligned} \alpha _0= & {} (B_{2R}(\theta )-B_0)^2-(B_{1R}(\theta )-B_{3I}(\theta ))^2,\\ \alpha _1= & {} 2(B_{1R}(\theta )+B_{3R}(\theta ))(B_{2R}(\theta )-B_0),\\ \alpha _2= & {} (B_{1R}(\theta )+B_{3R}(\theta ))^2+4B_0(B_{2R}(\theta )-B_0) \\&+\,(B_{1I}(\theta )-B_{3I}(\theta ))^2,\\ \alpha _3= & {} 4B_0(B_{1R}(\theta )+B_{3R}(\theta )),\\ \alpha _4= & {} 4B_0^2. \end{aligned}$$

Appendix D

$$\begin{aligned} \beta _0= & {} u_1^*u_2^*a_1a_2a_3-u_1^*u_2^*u_3^*(c_{13}c_{31}a_2 \\&+\,c_{12}c_{21}a_3+c_{23}c_{32}a_1) \\&+\,u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31} +c_{13}c_{21}c_{32}),\\ \beta _1= & {} u_1^*u_2^*a_1a_2+u_2^*u_3^*a_2a_3 \\&+\,u_1^*u_3^*a_1a_3-u_1^*u_3^*c_{13}c_{31} \\&-\,u_1^*u_2^*c_{12}c_{21}-u_2^*u_3^*c_{23}c_{32},\\ \beta _2= & {} u_2^*a_2+u_1^*a_1+u_3^*a_3. \end{aligned}$$

Appendix E

$$\begin{aligned} G_1= & {} \cos \theta _0\varrho _0\left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\cos \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\cos \frac{(p_2-1)\pi }{2}\right] \\&+\,\sin \theta _0\varrho _0\left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\sin \frac{(p_3-1)\pi }{2}\right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\sin \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\sin \frac{(p_2-1)\pi }{2}\right] \\&+\,u_2^*a_2(p_1+p_3)\theta _0^{p_1+p_3-1}\cos \frac{(p_1+p_3-1)\pi }{2} \\&+u_1^*a_1(p_2+p_3)\theta _0^{p_2+p_3-1}\cos \frac{(p_2+p_3-1)\pi }{2}\\&+\,u_3^*a_3(p_1+p_2)\theta _0^{p_1+p_2-1}\cos \frac{(p_1+p_2-1)\pi }{2} \\&+\,\cos \theta _0\varrho _0\left[ (p_1+p_2+p_3)\theta _0^{p_1+p_2+p_3-1}\right. \\&\times \cos \frac{(p_1+p_2+p_3-1)\pi }{2} \\&-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \cos \frac{(p_2-1)\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\\&\left. -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\right] -\sin \theta _0\varrho _0\\&\times \left[ (p_1+p_2+p_3)\theta _0^{p_1+p_2+p_3-1} \right. \\&\times \sin \frac{(p_1+p_2+p_3-1)\pi }{2}\\&-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \sin \frac{(p_2-1)\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _0^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\sin \frac{(p_1-1)\pi }{2}\\&\left. -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\sin \frac{(p_1-1)\pi }{2}\right] , \end{aligned}$$
$$\begin{aligned} G_2= & {} \cos \theta _0\varrho _0 \\&\left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\sin \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\sin \frac{(p_2-1)\pi }{2}\right] \\&-\,\sin \theta _0\varrho _0 \\&\left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\cos \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\cos \frac{(p_2-1)\pi }{2}\right] \\&+\,u_2^*a_2(p_1+p_3)\theta _0^{p_1+p_3-1}\sin \frac{(p_1+p_3-1)\pi }{2} \\&+\,u_1^*a_1(p_2+p_3)\theta _0^{p_2+p_3-1}\sin \frac{(p_2+p_3-1)\pi }{2}\\&+\,u_3^*a_3(p_1+p_2)\theta _0^{p_1+p_2-1} \\&\sin \frac{(p_1+p_2-1)\pi }{2}+\cos \theta _0\varrho _0\\&\times \left[ (p_1+p_2+p_3)\theta _0^{p_1+p_2+p_3-1} \right. \\&\times \sin \frac{(p_1+p_2+p_3-1)\pi }{2}\\&-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \sin \frac{(p_2-1)\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _0^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\sin \frac{(p_1-1)\pi }{2} \\&\left. -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\sin \frac{(p_1-1)\pi }{2}\right] \\&+\,\sin \theta _0\varrho _0\\&\times \left[ (p_1+p_2+p_3)\theta _0^{p_1+p_2+p_3-1} \right. \\&\cos \frac{(p_1+p_2+p_3-1)\pi }{2} \\&-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \cos \frac{(p_2-1)\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\\&\left. -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\right] , \end{aligned}$$
$$\begin{aligned} H_1= & {} \theta _0\sin \theta _0\varrho _0\left( u_1^*u_2^*a_1a_2\theta _0^{p_3}\cos \frac{p_3\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3\theta _0^{p_1}\cos \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _0^{p_2}\cos \frac{p_2\pi }{2}+1\right) \\&-\theta _0\cos \theta _0\varrho _0 \left( u_1^*u_2^*a_1a_2\theta _0^{p_3}\sin \frac{p_3\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3\theta _0^{p_1}\sin \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _0^{p_2}\sin \frac{p_2\pi }{2}\right) +2B_0\theta _0\sin 2\theta _0\varrho _0, \end{aligned}$$
$$\begin{aligned} H_2= & {} -\theta _0\sin \theta _0\varrho _0\left( u_1^*u_2^*a_1a_2\theta _0^{p_3}\sin \frac{p_3\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3\theta _0^{p_1}\sin \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _0^{p_2}\sin \frac{p_2\pi }{2}\right) +\theta _0\cos \theta _0\varrho _0 \\&\left( u_1^*u_2^*a_1a_2\theta _0^{p_3}\cos \frac{p_3\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3\theta _0^{p_1}\cos \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _0^{p_2}\cos \frac{p_2\pi }{2}\right) \\&+\,2B_0\theta _0\cos 2\theta _0\varrho _0-\theta _0\cos \theta _0\varrho _0. \end{aligned}$$

Appendix F

$$\begin{aligned} M_0= & {} u_1^*u_2^*a_1a_2a_3+k_1u_2^*u_3^*a_2a_3,\\ M_1(s)= & {} u_1^*u_2^*a_1a_2s^{p_3}+u_2^*u_3^*a_2a_3s^{p_1} \\&+\,u_1^*u_3^*a_1a_3s^{p_2}+u_3^*a_3k_1(s^{p_2}-u_2^*a_2),\\ M_2(s)= & {} u_2^*a_2s^{p_1+p_3}+u_1^*a_1s^{p_2+p_3}+u_3^*a_3s^{p_1+p_2} \\&-\,u_1^*u_2^*u_3^*(c_{13}c_{31}a_2+c_{12}c_{21}a_3+c_{23}c_{32}a_1)\\&+\,k_1[(s^{p_2}-u_2^*a_2)s^{p_3} \\&-\,u_3^*a_3s^{p_2}-u_2^*u_3^*c_{23}c_{32}],\\ M_3(s)= & {} s^{p_1+p_2+p_3} \\&+\,u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31}+c_{13}c_{21}c_{32}) \\&-\,u_1^*u_3^*c_{13}c_{31}s^{p_2} \\&-\,u_1^*u_2^*c_{12}c_{21}s^{p_3} \\&-\,u_2^*u_3^*c_{23}c_{32}s^{p_1}\\&-\,k_1s^{p_2+p_3}+k_1u_2^*u_3^*c_{23}c_{32}. \end{aligned}$$

Appendix G

$$\begin{aligned} M_{1R}(\theta )= & {} u_1^*u_2^*a_1a_2\theta ^{p_3}\cos \frac{p_3\pi }{2} \\&+\,u_2^*u_3^*a_2a_3\theta ^{p_1}\cos \frac{p_1\pi }{2} \\&+\,u_1^*u_3^*a_1a_3\theta ^{p_2}\cos \frac{p_2\pi }{2}\\&+\,u_3^*a_3k_1\left( \theta ^{p_2}\cos \frac{p_2\pi }{2}-u_2^*a_2\right) ,\\ M_{1I}(\theta )= & {} u_1^*u_2^*a_1a_2\theta ^{p_3}\sin \frac{p_3\pi }{2} \\&+\,u_2^*u_3^*a_2a_3\theta ^{p_1}\sin \frac{p_1\pi }{2} \\&+\,u_1^*u_3^*a_1a_3\theta ^{p_2}\sin \frac{p_2\pi }{2}\\&+\,u_3^*a_3k_1\theta ^{p_2}\sin \frac{p_2\pi }{2},\\ M_{2R}(\theta )= & {} u_2^*a_2\theta ^{p_1+p_3}\cos \frac{(p_1+p_3)\pi }{2} \\&+\,u_1^*a_1\theta ^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2}\\&+\,u_3^*a_3\theta ^{p_1+p_2}\cos \frac{(p_1+p_2)\pi }{2} \\&-\,u_1^*u_2^*u_3^*(c_{13}c_{31}a_2+c_{12}c_{21}a_3+c_{23}c_{32}a_1)\\&+\,k_1\left[ \theta ^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2} \right. \\&-\,u_2^*a_2\theta ^{p_3}\cos \frac{p_3\pi }{2}-u_3^*a_3\theta ^{p_2}\cos \frac{p_2\pi }{2} \\&\left. -\,u_2^*u_3^*c_{23}c_{32}\right] , \end{aligned}$$
$$\begin{aligned} M_{2I}(\theta )= & {} u_2^*a_2\theta ^{p_1+p_3}\sin \frac{(p_1+p_3)\pi }{2} \\&+\,u_1^*a_1\theta ^{p_2+p_3}\sin \frac{(p_2+p_3)\pi }{2} \\&+\,u_3^*a_3\theta ^{p_1+p_2}\sin \frac{(p_1+p_2)\pi }{2}\\&+\,k_1\left[ \theta ^{p_2+p_3}\sin \frac{(p_2+p_3)\pi }{2} \right. \\&-\,u_2^*a_2\theta ^{p_3}\sin \frac{p_3\pi }{2} \\&\left. -\,u_3^*a_3\theta ^{p_2}\sin \frac{p_2\pi }{2}\right] ,\\ M_{3R}(\theta )= & {} \theta ^{p_1+p_2+p_3}\cos \frac{(p_1+p_2+p_3)\pi }{2} \\&+\,u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31}+c_{13}c_{21}c_{32})\\&-\,u_1^*u_3^*c_{13}c_{31}\theta ^{p_2}\cos \frac{p_2\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}\theta ^{p_3}\cos \frac{p_3\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}\theta ^{p_1}\cos \frac{p_1\pi }{2}\\&-\,k_1\theta ^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2} \\&+\,k_1u_2^*u_3^*c_{23}c_{32},\\ M_{3I}(\theta )= & {} \theta ^{p_1+p_2+p_3}\cos \frac{(p_1+p_2+p_3)\pi }{2} \\&-\,u_1^*u_3^*c_{13}c_{31}\theta ^{p_2}\sin \frac{p_2\pi }{2}\\&-\,u_1^*u_2^*c_{12}c_{21}\theta ^{p_3}\sin \frac{p_3\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}\theta ^{p_1}\sin \frac{p_1\pi }{2} \\&-\,k_1\theta ^{p_2+p_3}\sin \frac{(p_2+p_3)\pi }{2}. \end{aligned}$$

Appendix H

$$\begin{aligned} \gamma _0= & {} (M_{2R}(\theta )-M_0)^2 -(M_{1R}(\theta )-M_{3I}(\theta ))^2,\\ \gamma _1= & {} 2(M_{1R}(\theta )+M_{3R}(\theta ))(M_{2R}(\theta )-M_0),\\ \gamma _2= & {} (M_{1R}(\theta )+M_{3R}(\theta ))^2+4M_0(M_{2R}(\theta )-M_0) \\&+\,(M_{1I}(\theta )-M_{3I}(\theta ))^2,\\ \gamma _3= & {} 4M_0(B_{1R}(\theta )+M_{3R}(\theta )), \\ \gamma _4= & {} 4M_0^2. \end{aligned}$$

Appendix I

$$\begin{aligned} \delta _0= & {} u_1^*u_2^*a_1a_2a_3 \\&-\,u_1^*u_2^*u_3^*(c_{13}c_{31}a_2+c_{12}c_{21}a_3+c_{23}c_{32}a_1) \\&+\,u_1^*u_2^*u_3^*(c_{12}c_{21}c_{31}+c_{13}c_{21}c_{32}),\\ \delta _1= & {} u_1^*u_2^*a_1a_2+u_2^*u_3^*a_2a_3 \\&+\,u_1^*u_3^*a_1a_3-u_1^*u_3^*c_{13}c_{31}-u_1^*u_2^*c_{12}c_{21} \\&-\,u_2^*u_3^*c_{23}c_{32}-k_1u_2^*a_2,\\ \delta _2= & {} u_2^*a_2+u_1^*a_1+u_3^*a_3. \end{aligned}$$

Appendix J

$$\begin{aligned} P_1= & {} \cos \theta _{0*}\varrho _{0*} \\&\times \left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\cos \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\cos \frac{(p_2-1)\pi }{2}\right] \\&+\,\sin \theta _{0*}\varrho _{0*} \\&\times \left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta ^{p_1-1}\sin \frac{(p_1-1)\pi }{2} \\&\left. +u_1^*u_3^*a_1a_3p_2\theta ^{p_2-1}\sin \frac{(p_2-1)\pi }{2}\right] \\&+\,u_2^*a_2(p_1+p_3)\theta _{0*}^{p_1+p_3-1}\cos \frac{(p_1+p_3-1)\pi }{2} \\&+\,u_1^*a_1(p_2+p_3)\theta _{0*}^{p_2+p_3-1} \\&\cos \frac{(p_2+p_3-1)\pi }{2}\\&+\,u_3^*a_3(p_1+p_2)\theta _0^{p_1+p_2-1}\cos \frac{(p_1+p_2-1)\pi }{2} \\&+\,\cos \theta _{0*}\varrho _{0*}\\&\times \left[ (p_1+p_2+p_3)\theta _{0*}^{p_1+p_2+p_3-1} \right. \\&\times \cos \frac{(p_1+p_2+p_3-1)\pi }{2}\\&-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \cos \frac{(p_2-1)\pi }{2} \\&-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _{0*}^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \\&-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _{0*}^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\\&\left. -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\right] \\&-\,\sin \theta _{0*}\varrho _{0*} \end{aligned}$$
$$\begin{aligned}&\qquad \;\times \left[ (p_1+p_2+p_3)\theta _{0*}^{p_1+p_2+p_3-1} \right. \\&\qquad \;\times \sin \frac{(p_1+p_2+p_3-1)\pi }{2}\\&\qquad \;-\,u_1^*u_3^*c_{13}c_{31}p_2\theta _0^{p_2-1} \sin \frac{(p_2-1)\pi }{2} \\&\qquad \;-u_1^*u_2^*c_{12}c_{21}p_3\theta _{0*}^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \\&\qquad \; -\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\sin \frac{(p_1-1)\pi }{2}\\&\left. \qquad \;-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _{0*}^{p_1-1}\sin \frac{(p_1-1)\pi }{2}\right] \\&\qquad \;+\,k_1(p_2+p_3)\theta _{0*}^{p_2+p_3-1} \\&\qquad \;\times \cos \frac{(p_2+p_3-1)\pi }{2}\\&\qquad \;+\, u_3^*a_3k_1p_2\theta _{0*}^{p_2-1} \\&\qquad \;\times \left[ \cos \frac{(p_2-1)\pi }{2}\cos \theta _{0*}\varrho _{0*} \right. \\&\left. \qquad \;+\,\sin \frac{(p_2-1)\pi }{2}\sin \theta _{0*}\varrho _{0*}\right] \\&\qquad \;-\,u_2^*a_2k_1p_3\theta _{0*}^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \\&\qquad \;-u_3^*a_3k_1p_2\theta _{0*}^{p_2-1}\cos \frac{(p_2-1)\pi }{2} \\&\qquad \;-\,k_1(p_2+p_3)\theta _{0*}^{p_2+p_3-1} \\&\qquad \;\times \left[ \cos \frac{(p_2+p_3-1)\pi }{2}\cos \theta _{0*}\varrho _{0*} \right. \\&\left. \qquad \;-\,\sin \frac{(p_2+p_3-1)\pi }{2}\sin \theta _{0*}\varrho _{0*} \right] , \end{aligned}$$
$$\begin{aligned} P_2= & {} \cos \theta _{0*}\varrho _{0*} \\&\left[ u_1^*u_2^*a_1a_2p_3\theta _{0*}^{p_3-1} \right. \\&\sin \frac{(p_3-1)\pi }{2}+u_2^*u_3^*a_2a_3p_1\theta _{0*}^{p_1-1} \\&\sin \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta _{0*}^{p_2-1}\sin \frac{(p_2-1)\pi }{2}\right] \\&-\,\sin \theta _{0*}\varrho _{0*} \\&\left[ u_1^*u_2^*a_1a_2p_3\theta _0^{p_3-1}\cos \frac{(p_3-1)\pi }{2}\right. \\&+\,u_2^*u_3^*a_2a_3p_1\theta _{0*}^{p_1-1}\cos \frac{(p_1-1)\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3p_2\theta _{0*}^{p_2-1}\cos \frac{(p_2-1)\pi }{2}\right] \end{aligned}$$
$$\begin{aligned}&\qquad \;+\,u_2^*a_2(p_1+p_3)\theta _{0*}^{p_1+p_3-1}\sin \frac{(p_1+p_3-1)\pi }{2} \\&\qquad \; +\,u_1^*a_1(p_2+p_3)\theta _{0*}^{p_2+p_3-1}\sin \frac{(p_2+p_3-1)\pi }{2}\\&\qquad \;+\,u_3^*a_3(p_1+p_2)\theta _{0*}^{p_1+p_2-1}\sin \frac{(p_1+p_2-1)\pi }{2} \\&\qquad \; +\,\cos \theta _{0*}\varrho _{0*}\\&\qquad \;\times \left[ (p_1+p_2+p_3)\theta _{0*}^{p_1+p_2+p_3-1} \right. \\&\qquad \;\times \sin \frac{(p_1+p_2+p_3-1)\pi }{2} \\&\qquad \; -\,u_1^*u_3^*c_{13}c_{31}p_2\theta _{0*}^{p_2-1} \sin \frac{(p_2-1)\pi }{2} \\&\qquad \;-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _{0*}^{p_3-1} \\&\qquad \;\times \sin \frac{(p_3-1)\pi }{2}-u_2^*u_3^*c_{23}c_{32}p_1\theta _{0*}^{p_1-1} \\&\qquad \;\times \sin \frac{(p_1-1)\pi }{2}\\&\qquad \;-u_2^*u_3^*c_{23}c_{32}p_1\theta _{0*}^{p_1-1} \\&\left. \qquad \;\times \sin \frac{(p_1-1)\pi }{2}\right] +\sin \theta _{0*}\varrho _{0*} \\&\qquad \;\times \left[ (p_1+p_2+p_3)\theta _{0*}^{p_1+p_2+p_3-1} \right. \\&\qquad \;\times \cos \frac{(p_1+p_2+p_3-1)\pi }{2}\\&\qquad \;-u_1^*u_3^*c_{13}c_{31}p_2\theta _{0*}^{p_2-1} \cos \frac{(p_2-1)\pi }{2} \\&\qquad \;-\,u_1^*u_2^*c_{12}c_{21}p_3\theta _{0*}^{p_3-1}\cos \frac{(p_3-1)\pi }{2} \\&\qquad \;-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _0^{p_1-1}\cos \frac{(p_1-1)\pi }{2} \\&\left. \qquad \;-\,u_2^*u_3^*c_{23}c_{32}p_1\theta _{0*}^{p_1-1}\cos \frac{(p_1-1)\pi }{2}\right] \\&\qquad \;+\,k_1(p_2+p_3)\theta _{0*}^{p_2+p_3-1} \\&\qquad \;\times \sin \frac{(p_2+p_3-1)\pi }{2}\\&\qquad \;+\, u_3^*a_3k_1p_2\theta _{0*}^{p_2-1} \\&\qquad \;\times \left[ \sin \frac{(p_2-1)\pi }{2}\cos \theta _{0*}\varrho _{0*} \right. \\&\left. \qquad \;+\,\cos \frac{(p_2-1)\pi }{2}\sin \theta _{0*}\varrho _{0*}\right] \\&\qquad \;-\,u_2^*a_2k_1p_3\theta _{0*}^{p_3-1}\sin \frac{(p_3-1)\pi }{2} \\&\qquad \; -\,u_3^*a_3k_1p_2\theta _{0*}^{p_2-1}\sin \frac{(p_2-1)\pi }{2} \\&\qquad \;-\,k_1(p_2+p_3)\theta _0^{p_2+p_3-1} \\&\qquad \;\times \left[ \sin \frac{(p_2+p_3-1)\pi }{2}\cos \theta _0\varrho _0 \right. \\&\left. \qquad \;-\,\cos \frac{(p_2+p_3-1)\pi }{2}\sin \theta _0\varrho _0 \right] , \end{aligned}$$
$$\begin{aligned} Q_1= & {} \theta _{0*}\sin \theta _{0*}\varrho _{0*}\left( u_1^*u_2^*a_1a_2\theta _{0*}^{p_3} \right. \\&\times \cos \frac{p_3\pi }{2}+u_2^*u_3^*a_2a_3\theta _{0*}^{p_1}\cos \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _0^{p_2}\cos \frac{p_2\pi }{2}+1\right) \\&-\,\theta _{0*}\cos \theta _{0*}\varrho _{0*} \\&\times \left( u_1^*u_2^*a_1a_2\theta _{0*}^{p_3}\sin \frac{p_3\pi }{2}+u_2^*u_3^*a_2a_3\theta _{0*}^{p_1} \right. \\&\left. \sin \frac{p_1\pi }{2} +u_1^*u_3^*a_1a_3\theta _{0*}^{p_2}\sin \frac{p_2\pi }{2}\right) \\&+2B_0\theta _{0*}\sin 2\theta _{0*}\varrho _{0*}+u_3^*a_3k_1 \\&\left[ \cos \theta _{0*}\varrho _{0*} \left( \theta _{0*}^{p_2}\cos \frac{p_2\pi }{2}-u_2^*a_2\right) \right. \\&\left. +\,\theta _{0*}^{p_2}\sin \theta _{0*}\varrho _{0*}\sin \frac{p_2\pi }{2} \right] \\&+\,k_1\cos \theta _{0*}\varrho _{0*} \\&\left[ \theta _{0*}^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2} \right. \\&\left. +u_2*u_3^*c_{23}c_{32}\right] \\&-\,k_1\theta _{0*}^{p_2+p_3}\sin \theta _{0*}\varrho _{0*} \sin \frac{(p_2+p_3)\pi }{2}, \end{aligned}$$
$$\begin{aligned} Q_2= & {} -\theta _{0*}\sin \theta _{0*}\varrho _{0*}\left( u_1^*u_2^*a_1a_2\theta _{0*}^{p_3}\sin \frac{p_3\pi }{2} \right. \\&+\,u_2^*u_3^*a_2a_3\theta _{0*}^{p_1}\sin \frac{p_1\pi }{2} \\&\left. +\,u_1^*u_3^*a_1a_3\theta _{0*}^{p_2}\sin \frac{p_2\pi }{2}\right) \\&+\,\theta _0\cos \theta _{0*}\varrho _{0*} \\&\times \left( u_1^*u_2^*a_1a_2\theta _{0*}^{p_3} \cos \frac{p_3\pi }{2}\right. \\&+u_2^*u_3^*a_2a_3\theta _{0*}^{p_1}\cos \frac{p_1\pi }{2} \\&\left. +u_1^*u_3^*a_1a_3\theta _{0*}^{p_2}\cos \frac{p_2\pi }{2}\right) \\&+\,2B_0\theta _{0*}\cos 2\theta _{0*}\varrho _{0*} \\&-\,\theta _{0*}\cos \theta _{0*}\varrho _{0*} \\&+\,u_3^*a_3k_1\left[ \cos \theta _{0*}\varrho _{0*} \sin \frac{p_2\pi }{2}\right. \\&\left. -\,\sin \theta _{0*}\varrho _{0*}\left( \theta _{0*}^{p_2}\cos \frac{p_2\pi }{2} -\,u_2^*a_2\right) \right] \\&+\,k_1\theta _{0*}^{p_2+p_3}\cos \theta _{0*}\varrho _{0*} \sin \frac{(p_2+p_3)\pi }{2}\\&+\,k_1\sin \theta _{0*}\varrho _{0*}\\&\times \,\left[ \theta _{0*}^{p_2+p_3}\cos \frac{(p_2+p_3)\pi }{2} -\,u_2*u_3^*c_{23}c_{32}\right] . \end{aligned}$$

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Xu, C., Liao, M. & Li, P. Bifurcation control for a fractional-order competition model of Internet with delays. Nonlinear Dyn 95, 3335–3356 (2019). https://doi.org/10.1007/s11071-018-04758-w

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Keywords

  • Bifurcation control
  • Competitive web site model
  • Internet
  • Stability
  • Hopf bifurcation
  • Fractional order
  • Delay