A simple family of exceptional maps with chaotic behavior (Q2301850)

In this paper chaotic properties of a simple family of maps in \(\mathbb{T}^2=\mathbb{R}^2\) (mod 1) are presented. For a deterministic map \(F\) defined of a compact set \(K\subset\mathbb{R}^n\), the following conditions that characterize the chaos are considered: \(\bullet\) Sensitive dependence to initial conditions (SDIC). A point \(x\in K\) is said to have SDIC if it exists a fixed \(\alpha > 0\) such that for every \(\varepsilon > 0\) there exists a point \(y\in B_\varepsilon(x)\cap K\) and a value \(N = N(\varepsilon, x, y)\in\mathbb{N}\) such that \(d(F^N(y), F^N(x))>\alpha,\) where \(d\) denotes the distance in \(K.\) This implies lack of predictability. We shall require SDIC for all \(x\in K.\) \(\bullet\) Topological transitivity (TT). A map \(F:K \to K\) is said to be TT if for every couple of open sets \(A, B \subset K,\) it exists \(m > 0\) such that \(F^m(A)\cap B\not =\emptyset.\) In Section 2, a model on \(\mathbb{T}^2\) is introduced, namely \[F_{\varepsilon,\omega}\left(\begin{array}{c}x\\y\end{array}\right):=\left(\begin{array}{c}x+\omega f_\varepsilon(x,y)\\y+f_\varepsilon(x,y)\end{array}\right),\] where \(\omega\) is an irrational number, \(\varepsilon>0\) is a small parameter, and \[f_\varepsilon(x,y)=\varepsilon (\sin^2(\pi x)+\sin^2(\pi y)).\] Furthermore, it is proved that it satisfies the SDIC and TT conditions. Some statistical properties are presented. In Section 3 a generalization is considered. Statistical properties, some bounds, and some additional numerical results are given. In Section 4, some possible extensions are mentioned.