Chaotic orbits for differentiable maps near anti-integrable limits

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• 作者：Te-Hsuan Chen^a ; ^{tsyoung.am97g@nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Wen-Wei Lin^a ; ^{wwlin@math.nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Chen-Chang Peng^b ; ^{ccpeng@mail.ncyu.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Anti-integrable limits ; Snap-back repellers ; Heteroclinical repellers ; Transversal homoclinic orbits ; Hé ; non maps ; The Arneodo&ndash ; Coullet&ndash ; Tresser maps
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：889-916
• 全文大小：494 K

文摘

In this work, we study the dynamical system which can be transformed into a difference equation of the form ϵg(x_t−n2,…,x_t,…,x_t+n1;ϵ)+g₀(x_t)=0$ϵg(x_{t-n_2},\dots ,x_t,\dots ,x_{t+n_1};ϵ)+g_0(x_t)=0$ where ϵ∈R$ϵ\in \mathbb{R}$ is a parameter, g:R^n₁+n₂+1→R$g:{\mathbb{R}}^{n_1+n_2+1}\to \mathbb{R}$ and g₀:R→R$g_0:\mathbb{R}\to \mathbb{R}$ are both C¹$C^1$ functions and n₁,n₂$n_1,n_2$ are both nonnegative integers. Suppose that the function g₀$g_0$ has k   simple zeros where k≥2$k\ge 2$. We give criteria of existence of some kinds of chaotic orbits, and construct infinite ones explicitly. As applications of these results, we establish snap-back repellers and heteroclinical repellers for the modified Mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the Hénon maps, the Arneodo–Coullet–Tresser maps and the quadratic volume preserving maps.