Multistability and cyclic attractors in duopoly games

A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T : …x; y† ! …r1…y†; r2…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† ˆ …r1 r2†…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 r1 and r2 are logistic maps. Ó 2000 Elsevier Science Ltd. All rights reserved.