A dynamic Cournot duopoly model with different strategies. (English) Zbl 1311.91040

Summary: This paper analyzes the dynamics of a Cournot duopoly model with different strategies. We offer results on existence, stability and local bifurcations of the equilibrium points. The bifurcation diagrams and Lyapunov exponents of the model are presented to show that the model behaves chaotically with the variation in the parameters. The state variables feedback and parameter variation methods are used to control the chaos of the model.


91A25 Dynamic games
91B54 Special types of economic markets (including Cournot, Bertrand)
91A05 2-person games
37G35 Dynamical aspects of attractors and their bifurcations
Full Text: DOI


[1] Cournot, A., Researches into the Mathematical Principles of the Theory of Wealth (1897), Macmillan: Macmillan USA
[2] Dixit, A., Comparative statics for oligopoly, Internat. Econ. Rev., 27, 107-122 (1986) · Zbl 0584.90012
[3] Puu, T., Chaos in duopoly pricing, Chaos Solit. Fract., 1, 573-581 (1991) · Zbl 0754.90015
[4] Kopel, M., Simple and complex adjustment dynamics in Cournot duopoly models, Chaos Solit. Fract., 12, 2031-2048 (1996) · Zbl 1080.91541
[5] Bischi, G. I.; Naimzada, A., Global analysis of a dynamic duopoly game with bounded rationality, (Dynamics Games and Application, vol. 5 (1999), Birkhouser), (Chapter 20)
[6] Bischi, G. I.; Stefanini, L.; Gardini, L., Synchronization intermittency and critical curves in a duopoly game, Math. Comput. Simul., 44, 559-585 (1998) · Zbl 1017.91500
[7] Agiza, H. N.; Bischi, G. I.; Kopel, M., Multistability in a dynamic Cournot game with three oligopolists, Math. Comput. Simul., 51, 63-90 (1999)
[8] Agiza, H. N.; Hegazi, A. S.; Elsadany, A. A., The dynamics of Bowleys model with bounded rationality, Chaos Solit. Fract., 12, 9, 1705-1717 (2001) · Zbl 1036.91004
[9] Ahmed, E.; Agiza, H. N.; Hassan, S. Z., On modifications of Puus dynamical duopoly, Chaos Solit. Fract., 11, 1025-1028 (2000) · Zbl 0955.91045
[10] Bischi, G. I.; Kopel, M., Equilibrium selection in a nonlinear duopoly game with adaptive expectations, J. Econ. Behav. Organ., 46, 73-100 (2001)
[11] Leonard, D.; Nishimura, K., Nonlinear dynamics in the Cournot model without full information, Ann. Oper. Res., 89, 165-173 (1999) · Zbl 0939.91096
[12] Den-Haan, W. J., The importance of the number of different agents in a heterogeneous asset pricing model, J. Econ. Dyn. Control, 25, 721746 (2001) · Zbl 0963.91051
[13] Agiza, H. N.; Hegazi, A. S.; Elsadany, A. A., Complex dynamics and synchronization of a duopoly game with bounded rationality, Math. Comput. Simul., 58, 133-146 (2002) · Zbl 1002.91010
[14] Agiza, H. N.; Elsadany, A. A., Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A, 320, 512-524 (2003) · Zbl 1010.91006
[15] Agiza, H. N.; Elsadany, A. A., Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput., 149, 843-860 (2004) · Zbl 1064.91027
[16] Zhang, J.; Da, Q.; Wang, Y., Analysis of nonlinear duopoly game with heterogeneous players, Econ. Modell., 24, 138-148 (2007)
[17] Angelini, N.; Dieci, R.; Nardini, F., Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules, Math. Comput. Simul., 79, 3179-3196 (2009) · Zbl 1169.91347
[18] Tramontana, F., Heterogeneous duopoly with isoelastic demand function, Econ. Modell., 27, 350-357 (2010)
[19] Naimzada, A.; Sbragia, L., Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes, Chaos Solit. Fract., 29, 707-722 (2006) · Zbl 1142.91340
[20] Mu, L.; Ma, J.; Liwen, C., A 3-dimensional discrete model of housing price and it inherent complexity analysis, J. Syst. Sci. Complexity, 22, 415-421 (2009) · Zbl 1193.93067
[21] Ma, J.; Mu, L., Dynamic analysis of the game between land supply and housing prices, Int. J. Comput. Math., 85, 983-992 (2008) · Zbl 1140.91323
[22] Elsadany, A. A., Competition analysis of a triopoly game with bounded rationality, Chaos Solit. Fract., 45, 1343-1348 (2012)
[24] Elaydi, S. N., An Introduction to Difference Equations (2005), Springer-Verlag · Zbl 1071.39001
[25] Fanti, L.; Gori, L., The dynamics of a differentiated duopoly with quantity competition, Econ. Modell., 29, 421-427 (2012)
[26] Elabbasy, E. M.; Agiza, H. N.; El-Metwally, H.; Elsadany, A. A., Bifurcation analysis, chaos and control in the Burgers mapping, Int. J. Nonlinear Sci., 4, 171-185 (2007) · Zbl 1394.37123
[27] Agiza, H. N., On the analysis of stability, bifurcation, chaos and chaos control of Kopel map, Chaos Solit. Fract., 10, 11, 909-1916 (1999) · Zbl 0955.37022
[28] Du, J.; Huang, T.; Sheng, Z., Analysis of decision-making in economic chaos control, Nonlinear Anal.: Real World Appl., 10, 2493-2501 (2009) · Zbl 1163.91331
[29] Holyst, J.; Urbanowicz, K., Chaos control in economical model by time-delayed feedback method, Phys. A: Stat. Mech. Appl., 287, 587-598 (2000)
[31] Elabbasy, E. M.; Agiza, H. N.; Elsadany, A. A., Analysis of nonlinear triopoly game with heterogeneous players, Comput. Math. Appl., 57, 488-499 (2009) · Zbl 1165.91324
[32] Ding, Z.; Hang, Q.; Yang, H., Analysis of the dynamics of multi-team Bertrand game with heterogeneous players, Int. J. Syst. Sci., 42, 047-1056 (2010) · Zbl 1214.91082
[33] Luo, X. S.; Chen, G. R.; Wang, B. H.; Fang, J. Q.; Zou, Y. L.; Quan, H. J., Control of period-doubling bifurcation and chaos in a discrete nonlinear system by the feedback of states and parameter adjustment, Acta. Phys. Sin.-Ch Ed, 52, 790-794 (2003)
[34] Pu, X.; Ma, J., Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations, Int. J. Bifurcat. Chaos, 23, 3, 1350053 (2013), (15 page) · Zbl 1270.39014
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