Multistability and cyclic attractors in duopoly games

A dynamic Cournot duopoly game, whose time evolution is modeled by the iter ation of a map T: (x,y) --> (r(1)(y), r(2)(x)), is considered. Results on t he existence of cycles and more complex attractors are given, based on the study of the one-dimensional map F(x) = (r(1) circle 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 pro perty 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 & Fract als, 7 (12) (1996) 2031-2048] as a model of an economic system, in which th e reaction functions r(1) and r(2) are logistic maps. (C) 2000 Elsevier Sci ence Ltd. All rights reserved.