Discrete processes with dynamical and semidynamical perturbations
Let X be a compact space and \(S: X\to X\) be a homeomorphism. Let \(A: X\to L({\mathbb{R}}^ d,{\mathbb{R}}^ d)\), \(\xi\) : \(X\to {\mathbb{R}}^ d\) and \(g: {\mathbb{R}}^ d\to {\mathbb{R}}^ d\) be continuous functions. For a fixed point \(x\in X\), the difference equation \[ u(n+1)=A(S^ n(x))g(u(n))+\xi (S^ n(x))\quad (n\in {\mathbb{Z}}) \] is considered and the existence of its solutions synchronous with the motion \(S^ n(x)\) is investigated. \(\{\) Reviewer's remark: Let \(Y=X\times {\mathbb{R}}^ d\), \(p(x,u)=x\), \(F(x,u)=(S(x),A(x)g(u)+\xi (x))\). Then p: (Y,F)\(\to (X,S)\) is an extension of discrete dynamical systems [see the reviewer's book ``Extensions of minimal transformation groups'' (1979; Zbl 0431.54023)]\(\}\).
scientific article; zbMATH DE number 4046672