Sensitivity, Shadowing Property and P-Chaos in Duopoly Games

In this paper, we discussed the cofinite sensitivity, shadowing property (SP), P-chaos, and chain mixing of a system induced by symmetric maps (Cournot maps) D(a,b)=(t(b),s(a)) over a product space G×H, where s:G→H, t:H→G, a∈G, b∈H, G and H are closed subintervals with G,H⊂R. The following hold: (1) D is cofinitely sensitive and equivalent to D2|Γ1 or D2|Γ2 being sensitive, where Γ1={(t(b),b):b∈H}, Γ2={(a,s(a)):a∈G}. (2) D possessing an SP is equivalent to both s∘t and t∘s having an SP. (3) t∘s possesses an SP if and only if s∘t does as well. (4) D is P-chaotic and equivalent to the maps s∘t and t∘s being P-chaotic. (5) If D is chain mixing, then both D2|Γ1 and D2|Γ2 are chain mixing. (6) If D2|Γ1 and D2|Γ2 are chain mixing, then D is chain transitive. Moreover, we extended (1)–(4) to three-dimensional cases.