Bifurcation control in the delayed fractional competitive web-site model with incommensurate-order
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ORIGINAL ARTICLE
Bifurcation control in the delayed fractional competitive web‑site model with incommensurate‑order Lingzhi Zhao1 · Jinde Cao2,3 · Chengdai Huang4 · Min Xiao5 · Ahmed Alsaedi3 · Bashir Ahmad3
Received: 9 September 2016 / Accepted: 31 July 2017 © Springer-Verlag GmbH Germany 2017
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. Keywords Competitive web-site · Time delay · Hopf bifurcation · Incommensurate-order · Bifurcation control
* Lingzhi Zhao [email protected] 1
School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China
2
Research Center for Complex Systems and Network Sciences, and School of Mathematics, Southeast University, Nanjing 210096, China
3
Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4
School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang 441053, China
5
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
1 Introduction “Web Wars: Which are the Most Valuable Weapons.” [1]. With the rapid development of network, Internet has become the necessity of people’s life and various web sites have sprung up. Meanwhile, the competition between similar web sites has become more intense and fiercer. In the past few years, many models of the competition dynamics of Internet, and other phenomena related to the World Wide Web have emerged [2–4]. The emergence of an information era mediated by the Internet brings about a great many novel and interesting economic problems. Network can be seen as a global market, web site maintenance to provide information to the user, and the number of visitors to the site in a period of time can become a successful proxy site. It is important to extract the competition structure of the sites from observed data on fluctuations in numbers of visitors to the sites, since such knowledge can help site maintainers plan their business strategies. Therefore, it can be necessary to incorporate competition structure to model the competitive dynamics of sites. More recently, various mathematical models have been proposed in the study of population dynamics [5, 6]. One of the famous models for dynamics of population is the Lotka–Volterra competition system, competition is an impor
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