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

  • Lingzhi ZhaoEmail author
  • Jinde Cao
  • Chengdai Huang
  • Min Xiao
  • Ahmed Alsaedi
  • Bashir Ahmad
Original Article


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.


Competitive web-site Time delay Hopf bifurcation Incommensurate-order Bifurcation control 



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.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Lingzhi Zhao
    • 1
    Email author
  • Jinde Cao
    • 2
    • 3
  • Chengdai Huang
    • 4
  • Min Xiao
    • 5
  • Ahmed Alsaedi
    • 3
  • Bashir Ahmad
    • 3
  1. 1.School of Information EngineeringNanjing Xiaozhuang UniversityNanjingChina
  2. 2.Research Center for Complex Systems and Network Sciences, and School of MathematicsSoutheast UniversityNanjingChina
  3. 3.Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  4. 4.School of Mathematics and Computer ScienceHubei University of Arts and ScienceXiangyangChina
  5. 5.College of AutomationNanjing University of Posts and TelecommunicationsNanjingChina

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