Abstract and Applied Analysis

Hopf Bifurcation Analysis in a Modified Price Differential Equation Model with Two Delays

Yanhui Zhai, Ying Xiong, and Xiaona Ma

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The paper investigates the behavior of price differential equation model based on economic theory with two delays. The primary aim of this thesis is to provide a research method to explore the undeveloped areas of the price model with two delays. Firstly, we modify the traditional price model by considering demand function as a downward opening quadratic function, and supply and demand functions both depending on the price of the past and the present. Then the price model with two delays is established. Secondly, by considering the price model with one delay, we get the stable interval. Regarding another delay as a parameter, we studied the linear stability and local Hopf bifurcation. In addition, we pay attention to the direction and stability of the bifurcating periodic solutions which are derived by using the normal form theory and center manifold method. Afterwards, the study turns to simulate the results through numerical analysis, which shows that the provided method is valid.

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Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 686274, 8 pages.

First available in Project Euclid: 6 October 2014

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Zhai, Yanhui; Xiong, Ying; Ma, Xiaona. Hopf Bifurcation Analysis in a Modified Price Differential Equation Model with Two Delays. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 686274, 8 pages. doi:10.1155/2014/686274. https://projecteuclid.org/euclid.aaa/1412607559

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  • 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.
  • S. Banerjee and W. A. Barnett, “Bifurcation analysis of Zellner's marshallain macroeconomic model,” Journal of Economic Dynamics and Control, vol. 35, no. 9, pp. 1577–1585.
  • L. Tanghong and L. Zhenwen, “Hopf bifurcation of price Reyleigh equation with time delay,” Journal of Jilin University, vol. 47, no. 3, 2009.
  • 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.
  • Y. Zhai, H. Bai, Y. Xiong, and X. Ma, “Hopf bifurcation analysis for the modified Rayleigh price model with time delay,” Abstract and Applied Analysis, vol. 2013, Article ID 290497, 6 pages, 2013.
  • O. I. Adeyemi and L. C. Hunt, “Modelling OECD industrial energy demand: asymmetric price responses and energy saving technical change,” Energy Economics, vol. 29, no. 4, pp. 693–709, 2007.
  • L. Tanghong and Z. Linhua, “Hopf and codimension two bifurcation for the price Rayleigh equation with two time delays,” Journal of Jilin University, vol. 50, no. 3, pp. 409–416, 2012.
  • K. L. Cooke and Z. Grossman, “Discrete delay, distributed delay and stability switches,” Journal of Mathematical Analysis and Applications, vol. 86, no. 2, pp. 592–627, 1982.
  • J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,” Physica D, vol. 130, no. 3-4, pp. 255–272, 1999.
  • J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.
  • D. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. \endinput