Multistability and cyclic attractors in duopoly games

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A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T:( x, y)→( r 1( y), r 2( x)), is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map F( x)=( r 1∘ r 2)( x). The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031–2048] as a model of an economic system, in which the reaction functions r 1 and r 2 are logistic maps.