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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 \[ \begin{aligned} x_{t+1} &= (1-\rho)x_t+ \rho\mu y_t(1- y_t),\\ y_{t+1} &= (1-\rho)y_t+ \rho\mu x_t(1-y_t) \end{aligned} \tag{1} \] where \(\rho,\mu\in \mathbb{R}_+\), \(x_t\) 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.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37N40 Dynamical systems in optimization and economics
37C75 Stability theory for smooth dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C35 Orbit growth in dynamical systems
65P30 Numerical bifurcation problems
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References:

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