Hopf Bifurcation and Stability Analysis in a Price Model with Time-Delayed Feedback International Journal of Mathematics Trends and Technology (IJMTT) © 2018 by IJMTT Journal Volume-62 Number-1 Year of Publication : 2018 Authors : Lei Peng, Yanhui Zhai 10.14445/22315373/IJMTT-V62P509 MLA Style: Lei Peng, Yanhui Zhai "Hopf Bifurcation and Stability Analysis in a Price Model with Time-Delayed Feedback" International Journal of Mathematics Trends and Technology 62.1 (2018): 55-66.

APA Style: Lei Peng, Yanhui Zhai (2018). Hopf Bifurcation and Stability Analysis in a Price Model with Time-Delayed Feedback. International Journal of Mathematics Trends and Technology, 62(1), 55-66.

Abstract
A Rayleigh price model with time-delayed feedback is investigated in this paper. First, a time-delayed feedback controller is introduced to the Rayleigh price model and we discussed the effect of the delay on the system. Second, the linear stability of the model and the local Hopf bifurcation are studied and we derived the conditions for the stability and the existence of Hopf bifurcation at the equilibrium of the system. Besides, the direction of Hopf bifurcation and the stability of bifurcation periodic solutions are studied by adopting the center manifold theorem and the normal form theory. At last, some numerical simulation results are confirmed that the feasibility of the theoretical analysis.

References
 Tang-Hong, L, and L. H. Zhou, “Hopf and Codimension Two Bifurcation in Price Rayleigh Equation with Two Time Delays,” Journal of Jilin University (Science Edition), vol. 50, no. 3, pp. 409-416, 2012.
 J. Li, W. Xu, W. Xie, Z. Ren, Research on nonlinear stochastic dynamical price model, Chaos Solitons & Fractals, vol. 37, no. 5, pp. 1391-1396, 2008.
 Tang-Hong, L, and L. H. Zhou, “Hopf and resonant codimension two bifurcation in price rayleigh equation with delays,” Journal of Northeast Normal University (Natrual Science Edition), vol. 44, no. 4, pp. 43–49,2012.
 W. Shuhe, Differential equation model and chaos, Journal of China Science and Technology University, pp. 312–324, 1999.
 Z. Xi-fan, C. Xia, and C. Yun-qing, “A qualitative analysis of price model in differential equations of price,” Journal of Shenyang Institute of Aeronautical Engineering, vol. 21, no. 1, pp. 83–86, 2004.
 T. Lv and Z. Liu, “Hopf bifurcation of price Rayleigh equation with time delay,” Journal of Jilin University, vol. 47, no. 3, pp. 441–448, 2009.
 B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
 Y. Kuang, Delay Differential Equations: With Applications in population Dynamics, Acdemic Press, New York, NY, USA, 1993.
 E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144-1165, 2002.
 J. Hale, Theory of Functional Differential Equations, Springer, 1977.
 W. Yong and Z. Yanhui, “Stability and Hopf bifurcation of differential equation model of price with time delay,” Highlights of Sciencepaper Online, vol. 4, no. 1, 2011.
 D. Tao, X. Liao, T. Huang, Dynamics of a congestion control model in a wireless access network, Nonlinear Analysis: Real World Applications, vol. 14, no .1, pp. 671-683, 2013.
 D. Ding, J. Zhu, X.S. Luo, Delay induced Hopf bifurcation in a dual model of Internet congestion, Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 2873-2883, 2009.
 Y.G. Zheng, Z.H. Wang, Stability and Hopf bifurcation of a class of TCP/AQM networks, Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1552-1559, 2010.
 Z.S. Cheng, J.D. Cao, Hybrid control of Hopf bifurcation in complex networks with delays, Neurocomputing, vol. 131, no. 131, pp. 164-170, 2014.
 D. Ding, X.Y. Zhang, et.al, Bifurcation control of complex networks model via PD controller, Neurocomputing ,vol. 175 (PA), pp. 1-9, 2016.
 D. Fan, J. Wei, Hopf bifurcation analysis in a tri-neuron network with time delay, Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 9-25,2008.
 S. Guo, H. Zheng, and Q. Liu, Hopf bifurcation analysis for congestion control with heterogeneous delays, Nonlinear Analysis: Real World Applications,vol. 11, no.4, pp. 3077-3090, 2010.

Keywords
Rayleigh price model,Time-delayed, Hopf bifurcation, Stability, Numerical simulation.