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Complexity analysis of a master-slave oligopoly model and chaos control. (English) Zbl 1406.91245

Summary: We establish a master-slave oligopoly game model with an upstream monopoly whose output is considered and two downstream oligopolies whose prices are considered. The existence and the local stable region of the Nash equilibrium point are investigated. The complex dynamic properties, such as bifurcation and chaos, are analyzed using bifurcation diagrams, the largest Lyapunov exponent diagrams, and the strange attractor graph. We further analyze the long-run average profit of the three firms and find that they are all optimal in the stable region. In addition, delay feedback control method and limiter control method are used in nondelayed model to control chaos. Furthermore, a delayed master-slave oligopoly game model is considered, and the three firms’ profit in various conditions is analyzed. We find that suitable delayed parameters are important for eliminating chaos and maximizing the profit of the players.

MSC:

91B54 Special types of economic markets (including Cournot, Bertrand)
91A80 Applications of game theory
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
34H10 Chaos control for problems involving ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations

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