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Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model. (English) Zbl 1232.91480
Summary: Complex nonlinear economic dynamics in a Cournot duopoly model proposed by M. Kopel is studied in detail in this work. By utilizing the topological horseshoe theory proposed by X. S. Yang, the authors detect the topological horseshoe chaotic dynamics in the Cournot duopoly model for the first time, and also give the rigorous computer-assisted verification for the existence of horseshoe. In the process of the proof, the topological entropy of the Cournot duopoly model is estimated to be bigger than zero, which implies that this economic system definitely exhibits chaos. In particular, the authors observe two different types of economic intermittencies, including the Pomeau-Manneville type-I intermittency arising near a saddle-node bifurcation, and the crisis-induced attractor widening intermittency caused by the interior crisis, which lead to the appearance of intermittency chaos. The authors also observe the transient chaos phenomenon which leads to the destruction of chaotic attractors. All these intermittency phenomena will help us to understand the similar dynamics observed in the practical stock market and the foreign exchange market. Besides, the Nash-equilibrium profits and the chaotic long-run average profits are analyzed. It is numerically demonstrated that both firms can have higher profits than the Nash-equilibrium profits, that is to say, both of the duopolists could be beneficial from a chaotic market. The controlled Cournot duopoly model can make one firm get more profit and reduce the profit of the other firm, and control the system to converge to an equilibrium state, where the two duopolists share the market equally.
MSC:
91B54Special types of economies
91B55Economic dynamics
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References:
[1] Day, R.: Irregular growth cycles. American economic review 72, 406-414 (1982)
[2] Boldrin, M.; Woodford, M.: Equilibrium models displaying endogenous fluctuations and chaos: a survey. Journal of monetary economics 25, 189-222 (1990)
[3] Puu, T.: Chaos in duopoly pricing. Chaos, solitons & fractals 1, 573-581 (1991) · Zbl 0754.90015
[4] Puu, T.: The chaotic duopolists revisited. Journal of economic behavior & organization 33, 385-394 (1998)
[5] Puu, T.: Complex dynamics with three oligopolists. Chaos, solitons & fractals 7, 2075-2081 (1996)
[6] Kopel, M.: Simple and complex adjustment dynamics in cournot duopoly models. Chaos, solitons & fractals 7, 2031-2048 (1996) · Zbl 1080.91541
[7] Qi, J.; Wang, D. W.; Ding, Y. S.; Wang, Z. D.: Dynamical analysis of a nonlinear competitive model with generic and brand advertising efforts. Nonlinear anal. Real world applications 8, 664-679 (2007) · Zbl 1152.34346
[8] Zhou, T. S.; Tang, Y.; Chen, G. R.: Chen’s attractor exists. International journal of bifurcation and chaos 14, 3167-3177 (2004) · Zbl 1129.37326
[9] Diallo, O.; Koné, Y.: Melnikov analysis of chaos in a general epidemiological model. Nonlinear analysis. Real world applications 8, 20-26 (2007) · Zbl 1114.37050
[10] Yang, X. S.; Tang, Y.: Horseshoe in piecewise continuous maps. Chaos, solitons & fractals 19, 841-845 (2004) · Zbl 1053.37006
[11] Yang, X. S.: Metric horseshoes. Chaos, solitons & fractals 20, 1149-1156 (2004) · Zbl 1048.37044
[12] Yang, X. S.; Li, Q. D.: Chaotic systems and chaotic circuits. (2007)
[13] Yang, X. S.; Yu, Y. G.; Zhang, S. C.: A new proof for existence of horseshoe in the Rössler system. Chaos, solitons & fractals 28, 223-227 (2003) · Zbl 1069.37030
[14] Yang, X. S.; Li, Q. D.: Horseshoe chaos in cellular neural networks. International journal of bifurcation and chaos 16, 157-161 (2006) · Zbl 1093.92009
[15] Pomeau, Y.; Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Communications in mathematical physics 74, 189-197 (1980)
[16] Grebogi, C.; Ott, E.; Romeiras, F.; Yorke, J. A.: Critical exponents for crisis-induced intermittency. Physical review A 36, 5365-5380 (1987)
[17] Mantegna, R. N.; Stanley, H. E.: Scaling behavior in the dynamics of an economic index. Nature 376, 46-49 (1995)
[18] Müller, U. A.; Dacorogna, M. M.; Olsen, R. B.; Pictet, O. V.; Schwarz, M.; Morgenegg, C.: Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis. Journal of banking & finance 14, 1189-1208 (1990)
[19] Chian, A. C. -L.; Rempel, E. L.; Rogers, C.: Crisis-induced intermittency in nonlinear economic cycles. Applied economics letters 14, 211-218 (2007)
[20] Chian, A. C. -L.; Rempel, E. L.; Rogers, C.: Complex economic dynamics: chaotic saddles, crisis and intermittency. Chaos, solitons & fractals 29, 1194-1218 (2006) · Zbl 1142.91652
[21] Chian, A. C. -L.; Rempel, E. L.; Vorotto, F. A.; Rogers, C.: Attractor merging crisis in chaotic business cycles. Chaos, solitons & fractals 24, 869-875 (2005) · Zbl 1081.37058
[22] Matsumoto, A.: Let it be: chaotic price instability can be beneficial. Chaos, solitons & fractals 18, 745-758 (2003) · Zbl 1069.37071
[23] Matsumoto, A.: Controlling the cournot--Nash chaos. Journal of optimization theory and applications 128, 379-392 (2006) · Zbl 1171.91331
[24] Agiza, H. N.: On the analysis of stability, bifurcation, chaos and chaos control of kopel map. Chaos, solitons & fractals 10, 1909-1916 (1999) · Zbl 0955.37022
[25] Govaerts, W.; Ghaziani, R. K.: Stable cycles in a cournot duopoly model of kopel. Journal of computational and applied mathematics 218, 247-258 (2008) · Zbl 1151.91458
[26] Xiong, J. C.: A note on topological entropy. Chinese science bulletin 34, 1673-1676 (1989) · Zbl 0691.54025
[27] Chen, L.; Chen, G. R.: Controlling chaos in an economic model. Physica A 374, 349-358 (2007)