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Frequency Domain Analysis for Hopf Bifurcation in a Delayed Competitive Web-site Model

Authors: Changjin Xu, Yusen Wu


In this paper, applying frequency domain approach, a delayed competitive web-site system is investigated. By choosing the parameter α as a bifurcation parameter, it is found that Hopf bifurcation occurs as the bifurcation parameter α passes a critical values. That is, a family of periodic solutions bifurcate from the equilibrium when the bifurcation parameter exceeds a critical value. Some numerical simulations are included to justify the theoretical analysis results. Finally, main conclusions are given.

Keywords: Stability, frequency domain, Nyquist criterion, Hopf bifurcation, Web-site system

Digital Object Identifier (DOI):

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[1] L.A. Adamic, B.A. Huberman, The nature of markets in the world wide web. Quart. J. Electronic Commence 1 (2000) 5-12.
[2] S.M. Maurer, B.A. Huberman, Competitive dynamics of web sites. J. Econ. Dynam. Control 27 (2003) 2195-2206.
[3] M. Xiao,J.D. Cao, Stability and Hopf bifurcation in a delayed competitive web sites model. Phys. Lett. A 353 (2006) 138-150.
[4] J.L. Zhang, J.H. Dou, Y. Shi, Hopf bifurcation of a competitive web-site system with reflexive and competition delays. Pure Appl. Math. 27 (2011) 51-54.(in chinese)
[5] D.J. Allwright, Harmonic balance and the Hopf bifurcation theorem. Math. Proc. Cambridge Phil. Soc. 82 (1977) 453-467.
[6] A.I. Mees,L.O. Chua, The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Trans. Circuits Syst. 26 (1979) 235-254.
[7] J.L. Moiola, G.R. Chen, Hopf bifurcation analysis: a frequency domain approach. Singapore: World Scientific, 1996.
[8] X.F. Liao, S.W. Li, Hopf bifurcation on a two-neuron system with distributed delays: a frequency domain approach. Nonlinear Dyn. 31 (2003) 299-326.
[9] X.F. Liao, S.W. Li,G.R. Chen, Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain. Neural Netw. 17 ( 2004) 545-561.
[10] J.L. Moiola, G.R. Chen, Frequency domain approach to computational analysis of bifurcations and limit cycles: a tutorial. Int. J. Bifur. Chaos 3 (1993) 843-867.
[11] C.J. Xu, X.H. Tang,M.X. Liao, Frequency domain analysis for bifur cation in a simplified tri-neuron BAM network model with two delays. Neural Netw. 23 (2010) 872-880.