Kato’s chaos in duopoly games

Let E,F⊂RE,F⊂R be two given closed intervals, and let τ: E → F and θ: F → E   be continuous maps. In this paper, we consider Koto’s chaos, sensitivity and accessibility of a given system Ψ(u,v)=(θ(v),τ(u))Ψ(u,v)=(θ(v),τ(u)) on a given product space E × F where u ∈ E and v ∈ F  . In particular, it is proved that for any Cournot map Ψ(u,v)=(θ(v),τ(u))Ψ(u,v)=(θ(v),τ(u)) on the product space E × F, the following hold: (1)If Ψ   satisfies Kato’s definition of chaos then at least one of Ψ2|Q1Ψ2|Q1 and Ψ2|Q2Ψ2|Q2 does, where Q1={(θ(v),v):v∈F}Q1={(θ(v),v):v∈F} and Q2={(u,τ(u)):u∈E}Q2={(u,τ(u)):u∈E}.(2)Suppose that Ψ2|Q1Ψ2|Q1 and Ψ2|Q2Ψ2|Q2 satisfy Kato’s definition of chaos, and that the maps θ and τ satisfy that for any ε > 0, if ∣n(τ∘θ)(v1)−n(τ∘θ)(v2)∣<ɛ∣(τ∘θ)n(v1)−(τ∘θ)n(v2)∣<ɛand ∣m(θ∘τ)(u1)−m(θ∘τ)(u2)∣<ɛ∣(θ∘τ)m(u1)−(θ∘τ)m(u2)∣<ɛfor some integers n, m > 0, then there is an integer l(n, m, ε) > 0 with ∣(τ∘θ)l(n,m,ɛ)(v1)−(τ∘θ)l(n,m,ɛ)(v2)∣<ɛ∣(τ∘θ)l(n,m,ɛ)(v1)−(τ∘θ)l(n,m,ɛ)(v2)∣<ɛand ∣(θ∘τ)l(n,m,ɛ)(u1)−(θ∘τ)l(n,m,ɛ)(u2)∣<ɛ.∣(θ∘τ)l(n,m,ɛ)(u1)−(θ∘τ)l(n,m,ɛ)(u2)∣<ɛ.Then Ψ satisfies Kato’s definition of chaos.