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On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. (English) Zbl 0955.37022
The author deals with the Cournot duopoly problem, that is $$\aligned x_{t+1} &= (1-\rho)x_t+ \rho\mu y_t(1- y_t),\\ y_{t+1} &= (1-\rho)y_t+ \rho\mu x_t(1-y_t) \endaligned \tag 1$$ where $\rho,\mu\in \bbfR_+$, $x_t$ and $y_t$ are production quantities. Here the author is interested only in positive solutions. He provides conditions for the stability of the fixed points and studies the bifurcation and chaos for (1), by computing the maximum Lyapunov exponents. Control of chaos is also discussed.

37E30Homeomorphisms and diffeomorphisms of planes and surfaces
37N40Dynamical systems in optimization and economics
37C75Stability theory
37D45Strange attractors, chaotic dynamics
37C35Orbit growth
65P30Bifurcation problems (numerical analysis)
Full Text: DOI
[1] Ahmed, E., Preprint. Mansoura University, 1997.
[2] Ahmed, E., Agiza, H. N., Hassan, S. M., On modelling advertisement in Cournot Duopoly. Chaos, Solitons and Fractals, 1999, 10(7), 1179--1184. · Zbl 0957.91066
[3] Devany, R. L., Introduction to Chaotic Dynamical Systems, Addison-Wesley, New York (1989).
[4] Edelstein-Keshet, L., Mathematical Models in Biology. Random House, New York, 1988. · Zbl 0674.92001
[5] Kaas, L., Stabilizing chaos in a dynamic macroeconomic model. Jour. of Economic Behaviour and Organization, 1999, (in press).
[6] Kopel, M., Simple and complex adjustment dynamics in Cournot Duopoly model. Chaos, Solitons and Fractals, 1996, 7(12), 2031--2048. · Zbl 1080.91541
[7] Luhta, I. and Virtanen, I., Chaos, Solitons and Fractals, 1996, 7, 2083.
[8] Ott, E., Grebogi, C., Yorke, J. A., Controlling chaos. Phys. Rev. Lett., 1990, 64, 1196--1199. · Zbl 0964.37501
[9] Sonis, M., Once more on Henon map: Analysis of bifurcations. Chaos, Solitons and Fractals, 1996, 7(12), 2215--2234. · Zbl 1080.91545