Advertisement

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

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 

Notes

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.

References

  1. 1.
    Strom D (1977) The best of push. Datamation 43(4):56–61Google Scholar
  2. 2.
    Adamic LA, Huberman BA (2000) The nature of markets in the world wide web. Q J Electron Commence 1:5–12Google Scholar
  3. 3.
    Maurer SM, Huberman BA (2003) Competitive dynamics of web sites. J Econ Dyn Control 27:2195–2206MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ren Y, Yang D, Diao X (2010) Websites competition model with market segmentation and its stability analysis. J Dalian Univ Technol 50(5):816–821MathSciNetGoogle Scholar
  5. 5.
    Cabo RM, Gimeno R (2013) Estimating population ecology models for the WWW market: evidence of competitive oligopolies. Nonlinear Dyn Psychol Life Sci 17(1):159–172Google Scholar
  6. 6.
    Aluja M, Ordano M, Guillen L, Rul J (2015) Understanding long-term fruit fly (Diptera: Tephritidae) population dynamics: implications for areawide management. J Econ Entomol 105(3):823–836CrossRefGoogle Scholar
  7. 7.
    Li J, Zhao A (2015) Stability analysis of a non-autonomous Lotka–Volterra competition model with seasonal succession. Appl Math Model 40(2):763–781MathSciNetCrossRefGoogle Scholar
  8. 8.
    Avelino PP, Bazeia D, Menezes J (2014) String networks in [formula omitted] Lotka–Volterra competition models. Phys Lett A 378(4):393–397CrossRefzbMATHGoogle Scholar
  9. 9.
    Jia Y, Wu J, Xu HK (2014) Positive solutions of a Lotka–Volterra competition model with cross-diffusion. Comput Math Appl 68(10):1220–1228MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent-II. Geophys J R Astron Soc 13:529–539CrossRefGoogle Scholar
  11. 11.
    Kvitsinskii AA (1993) Fractional integrals and derivatives: theory and applications. Teoret Mat Fiz 3:397–414Google Scholar
  12. 12.
    Sun HH, Abdelwahab AA, Onaral B (1984) Linear approximation of transfer function with a pole of fractional order. IEEE Trans Autom Control 29:441–444CrossRefzbMATHGoogle Scholar
  13. 13.
    Podlubny I (1999) Fractional differential equations. Academic Press, New YorkzbMATHGoogle Scholar
  14. 14.
    Mandelbrot BB (1982) The fractal geometry of nature. Henry Holt and Company, New YorkzbMATHGoogle Scholar
  15. 15.
    Rakkiyappan R, Cao JD, Velmurugan G (2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 1(26):84–97MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Liu H, Li S, Wang H, Huo Y, Luo J (2015) Adaptive synchronization for a class of uncertain fractional-order neural networks. Entropy 17(10):7185–7200MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li G, Liu H (2016) Stability analysis and synchronization for a class of fractional-order neural networks. Entropy 18(2):55CrossRefGoogle Scholar
  18. 18.
    Xiao M, Zheng WX, Jiang GP, Cao JD (2015) Undamped oscillations generated by hopf bifurcations in fractional-order recurrent neural networks with caputo derivative. IEEE Trans Neural Netw Learn Syst 26(12):3201–3214MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cao JD, Xiao M (2007) Stability and Hopf bifurcation in a simplified BAM neural network with two time delays. IEEE Trans Neural Netw 18:416–430CrossRefGoogle Scholar
  20. 20.
    Yu P (2004) Bifurcation dynamics in control systems. Bifurc Control 293(3):719–722Google Scholar
  21. 21.
    Abed EH, Fu JH (1987) Local feedback stabilization and bifurcation control: II. Stationary bifurcation. Syst Control Lett 8:467–473MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chen GR, Moiola JL, Wang HO (2000) Bifurcation control: theories, methods and applications. Int J Bifurc Chaos 10:511–548MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yu P, Chen G (2004) Hopf bifurcation control using nonlinear feedback with polynomial functions. Int J Bifurc Chaos 14:1683–1704MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pan Y, Yu H, Er MJ (2014) Adaptive neural pd control with semiglobal asymptotic stabilization guarantee. IEEE Trans Neural Netw Learn Syst 25(12):2264–2274CrossRefGoogle Scholar
  25. 25.
    Pan Y, Liu Y, Xu B, Yu H (2016) Hybrid feedback feedforward: an efficient design of adaptive neural network control. Neural Netw 76:122–134CrossRefGoogle Scholar
  26. 26.
    Pan Y, Yu H (2016) Composite learning from adaptive dynamic surface control. IEEE Trans Autom Control 61(9):2603–2609MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Xiao M, Ho DWC, Cao JD (2009) Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. Nonlinear Dyn 58:319–344MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shi M, Wang ZH (2013) Stability and Hopf bifurcation control of a fractional-order small world network model. Sci China Phys Mech 43(4):467–477Google Scholar
  29. 29.
    Min X, Cao J (2006) Stability and Hopf bifurcation in a delayed competitive web sites model. Phys Lett A 353(2–3):138–150zbMATHGoogle Scholar
  30. 30.
    Deng W, Li C, Lu J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang JL, Dou JH, Shi Y (2011) Hopf bifurcation of a competitive web-site system with reflexive and competition delays. Pure Appl Math 27:51–54MathSciNetzbMATHGoogle Scholar
  32. 32.
    Xu CJ, Wu YS (2015) Frequency domain analysis for Hopf bifurcation in a delayed competitive web-site model. Int J Comput Inf Sci Engine 9(2):138–141Google Scholar
  33. 33.
    Huang CD, Cao JD, Xiao M (2016) Hybrid control on bifurcation for a delayed fractional gene regulatory network. Chaos Solitons Fract 87:19–29MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang H, Yu Y, Wen G, Zhang S, Yu J (2015) Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154(C):15–23CrossRefGoogle Scholar
  35. 35.
    Abdelouahab MS, Hamri NE, Wang J (2012) Hopf bifurcation and chaos in fractional-order modified hybrid optical system. Nonlinear Dyn 69(1–2):275–284MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lozi RP, Abdelouahab MS (2015) Hopf Bifurcation and chaos in simplest fractional-order memristor-based electrical circuit. Indian J Ind Appl Math 6(2):105–119CrossRefGoogle Scholar
  37. 37.
    Padula F, Alcantara S, Vilanova R, Visioli A (2013) \(H_\infty\)control of fractional linear systems. Automatica 49:2276–2280MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Bhalekar S, Varsha D (2011) A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order. J Fract Calc Appl 1(5):1–9Google Scholar

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

Personalised recommendations