Hopf bifurcation analysis in a modified price differential equation model with two delays. (English) Zbl 1474.91098

Summary: 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.


91B55 Economic dynamics
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Shuhe, W., Differential equation model and chaos, Journal of China Science and Technology University, 312-324 (1999)
[2] Xi-fan, Z.; Xia, C.; Yun-qing, C., A qualitative analysis of price model in differential equations of price, Journal of Shenyang Institute of Aeronautical Engineering, 21, 1, 83-86 (2004)
[3] Banerjee, S.; Barnett, W. A., Bifurcation analysis of Zellner’s marshallain macroeconomic model, Journal of Economic Dynamics and Control, 35, 9, 1577-1585 · Zbl 1229.91221
[4] Tanghong, L.; Zhenwen, L., Hopf bifurcation of price Reyleigh equation with time delay, Journal of Jilin University, 47, 3 (2009)
[5] Yong, W.; Yanhui, Z., Stability and Hopf bifurcation of differential equation model of price with time delay, Highlights of Sciencepaper Online, 4, 1 (2011)
[6] Zhai, Y.; Bai, H.; Xiong, Y.; Ma, X., Hopf bifurcation analysis for the modified Rayleigh price model with time delay, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.91078 · doi:10.1155/2013/290497
[7] Adeyemi, O. I.; Hunt, L. C., Modelling OECD industrial energy demand: asymmetric price responses and energy saving technical change, Energy Economics, 29, 4, 693-709 (2007) · doi:10.1016/j.eneco.2007.01.007
[8] Tanghong, L.; Linhua, Z., Hopf and codimension two bifurcation for the price Rayleigh equation with two time delays, Journal of Jilin University, 50, 3, 409-416 (2012) · Zbl 1265.34299
[9] Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications, 86, 2, 592-627 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[10] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 3-4, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[11] Hale, J., Theory of Functional Differential Equations (1977), New York, NY, USA: Springer, New York, NY, USA · Zbl 0352.34001
[12] Hassard, D. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0474.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.