Kato’s chaos in duopoly games

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• 作者：Risong Li ; ^{gdoulrs@163.com" class="auth_mail" title="E-mail the corresponding author} ; ^{gdoulrs1@163.com" class="auth_mail" title="E-mail the corresponding author} ; Hongqing Wang ^{wanghq3333@126.com" class="auth_mail" title="E-mail the corresponding author} ; Yu Zhao ^{datom@189.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Sensitivity ; Accessibility ; Kato&rsquo ; s definition of chaos ; Duopoly game
• 刊名：Chaos, Solitons & Fractals
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：84
• 期：Complete
• 页码：69-72
• 全文大小：291 K

文摘

Let E,F⊂R$E,F\subset \mathbb{R}$ 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))$\Psi (u,v)=\left(\theta \right(v),\tau (u\left)\right)$ 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))$\Psi (u,v)=\left(\theta \right(v),\tau (u\left)\right)$ on the product space E × F, the following hold:

(1)

If Ψ   satisfies Kato’s definition of chaos then at least one of Ψ²|_Q1${\Psi }^2{|}_{Q_1}$ and Ψ²|_Q2${\Psi }^2{|}_{Q_2}$ does, where Q₁={(θ(v),v):v∈F}$Q_1=\{(\theta \left(v\right),v):v\in F\}$ and Q₂={(u,τ(u)):u∈E}$Q_2=\{(u,\tau \left(u\right)):u\in E\}$.

(2)

Suppose that Ψ²|_Q1${\Psi }^2{|}_{Q_1}$ and Ψ²|_Q2${\Psi }^2{|}_{Q_2}$ 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$\mid {(\tau \circ \theta )}^n\left(v_1\right)-{(\tau \circ \theta )}^n\left(v_2\right)\mid <\varepsilon$

and

1228bcd3d9880908fff4da0eae6e4368" title="Click to view the MathML source">∣m(θ∘τ)(u₁)−m(θ∘τ)(u₂)∣<ɛ$\mid {(\theta \circ \tau )}^m\left(u_1\right)-{(\theta \circ \tau )}^m\left(u_2\right)\mid <\varepsilon$

for some integers n, m > 0, then there is an integer l(n, m, ε) > 0 with

124f7560319fa543a5a595f2" title="Click to view the MathML source">∣(τ∘θ)^l(n,m,ɛ)(v₁)−(τ∘θ)^l(n,m,ɛ)(v₂)∣<ɛ$\mid {(\tau \circ \theta )}^{l(n,m,\varepsilon )}\left(v_1\right)-{(\tau \circ \theta )}^{l(n,m,\varepsilon )}\left(v_2\right)\mid <\varepsilon$

and

∣(θ∘τ)^l(n,m,ɛ)(u₁)−(θ∘τ)^l(n,m,ɛ)(u₂)∣<ɛ.$\mid {(\theta \circ \tau )}^{l(n,m,\varepsilon )}\left(u_1\right)-{(\theta \circ \tau )}^{l(n,m,\varepsilon )}\left(u_2\right)\mid <\varepsilon .$

Then Ψ satisfies Kato’s definition of chaos.