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Entropy 2016, 18(7), 266; doi:10.3390/e18070266

Complex Dynamics of a Continuous Bertrand Duopoly Game Model with Two-Stage Delay

1
College of Management and Economics, Tianjin University, Tianjin 300072, China
2
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
*
Author to whom correspondence should be addressed.
Academic Editor: J.A. Tenreiro Machado
Received: 13 May 2016 / Revised: 1 July 2016 / Accepted: 13 July 2016 / Published: 20 July 2016

Abstract

This paper studies a continuous Bertrand duopoly game model with two-stage delay. Our aim is to investigate the influence of delay and weight on the complex dynamic characteristics of the system. We obtain the bifurcation point of the system respect to delay parameter by calculating. In addition, the dynamic properties of the system are simulated by power spectrum, attractor, bifurcation diagram, the largest Lyapunov exponent, 3D surface chart, 4D Cubic Chart, 2D parameter bifurcation diagram, and 3D parameter bifurcation diagram. The results show that the stability of the system depends on the delay and weight, in order to maintain stability of price and ensure the firm profit, the firms must control the parameters in the reasonable region. Otherwise, the system will lose stability, and even into chaos, which will cause fluctuations in prices, the firms cannot be profitable. Finally, the chaos control of the system is carried out by a control strategy of the state variables’ feedback and parameter variation, which effectively avoid the damage of chaos to the economic system. Therefore, the results of this study have an important practical significance to make decisions with multi-stage delay for oligopoly firms. View Full-Text
Keywords: duopoly game; two-stage delay; bifurcation and chaos; dynamic characteristics duopoly game; two-stage delay; bifurcation and chaos; dynamic characteristics
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Ma, J.; Si, F. Complex Dynamics of a Continuous Bertrand Duopoly Game Model with Two-Stage Delay. Entropy 2016, 18, 266.

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