Spatio-temporal chaos in duopoly games

Suppose that G and H are two given closed subintervals of R, and that q : G→H and p : H→ G are continuous maps. Let Γ (s, t) = (p(t), q(s)) be a Cournot map over the space G ˟ H. In this paper, we study spatio-temporal chaos of such a Cournot map. In particular, it is shown that if p and q are onto maps, then the following are equivalent: 1) Γ is spatio-temporally chaotic; 2) Γ ²|Λ1 is spatio-temporally chaotic; 3) Γ ²|Λ2 is spatio-temporally chaotic; 4) Γ |Λ1 ∪Λ2 is spatio-temporally chaotic. Moreover, it is proved that if p and q are onto maps, then p ◦ q is spatio-temporally chaotic if and only if so is q ◦ p. Also, we give two examples which show that for the above results, it is necessary to assume that p and q are onto maps.