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 md5=0f42011ca0c8d19dfdbd853e054f1ee1" title="Click to view the MathML source">E,F⊂R

E, F \subset 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 md5=b0ed57eecc1f224a57dc8b397d89fb51" title="Click to view the MathML source">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 md5=b0ed57eecc1f224a57dc8b397d89fb51" title="Click to view the MathML source">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 md5=471ff4e177c75225713b1073f73afc58" title="Click to view the MathML source">Ψ²|_Q₁ $Ψ^{2} |_{Q_{1}}$ and md5=93f879fa1cb4c79827acb66d7a93807e" title="Click to view the MathML source">Ψ²|_Q₂ $Ψ^{2} |_{Q_{2}}$ does, where md5=233dbeacecbbfcf9af8823fdcb3066e1" title="Click to view the MathML source">Q₁={(θ(v),v):v∈F} $Q_{1} = {(θ (v), v) : v \in F}$ and md5=b55176639a994ea4b62fbba5172ee3f5" title="Click to view the MathML source">Q₂={(u,τ(u)):u∈E} $Q_{2} = {(u, τ (u)) : u \in E}$ .

(2)

Suppose that md5=471ff4e177c75225713b1073f73afc58" title="Click to view the MathML source">Ψ²|_Q₁ $Ψ^{2} |_{Q_{1}}$ and md5=93f879fa1cb4c79827acb66d7a93807e" title="Click to view the MathML source">Ψ²|_Q₂ $Ψ^{2} |_{Q_{2}}$ satisfy Kato’s definition of chaos, and that the maps θ and τ satisfy that for any ε > 0, if

md5=53466eecd59a4dbdce73edd8bb22de6b" title="Click to view the MathML source">∣n(τ∘θ)(v₁)−n(τ∘θ)(v₂)∣<ɛ

∣ {(τ \circ θ)}^{n} (v_{1}) - {(τ \circ θ)}^{n} (v_{2}) ∣ < ɛ

and

md5=1228bcd3d9880908fff4da0eae6e4368" title="Click to view the MathML source">∣m(θ∘τ)(u₁)−m(θ∘τ)(u₂)∣<ɛ

∣ {(θ \circ τ)}^{m} (u_{1}) - {(θ \circ τ)}^{m} (u_{2}) ∣ < ɛ

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

md5=d34cda2a124f7560319fa543a5a595f2" title="Click to view the MathML source">∣(τ∘θ)^l(n,m,ɛ)(v₁)−(τ∘θ)^l(n,m,ɛ)(v₂)∣<ɛ

∣ {(τ \circ θ)}^{l (n, m, ɛ)} (v_{1}) - {(τ \circ θ)}^{l (n, m, ɛ)} (v_{2}) ∣ < ɛ

and

md5=f036d16483503d6c6c134b58ac60245d" title="Click to view the MathML source">∣(θ∘τ)^l(n,m,ɛ)(u₁)−(θ∘τ)^l(n,m,ɛ)(u₂)∣<ɛ.

∣ {(θ \circ τ)}^{l (n, m, ɛ)} (u_{1}) - {(θ \circ τ)}^{l (n, m, ɛ)} (u_{2}) ∣ < ɛ .

Then Ψ satisfies Kato’s definition of chaos.

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