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

  • Changjin XuEmail author
  • Maoxin Liao
  • Peiluan Li
Original Paper


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.


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


Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Guizhou Key Laboratory of Economics System SimulationGuizhou University of Finance and EconomicsGuiyangPeople’s Republic of China
  2. 2.School of Mathematics and PhysicsUniversity of South ChinaHengyangPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangPeople’s Republic of China

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