Invariant tori of a class of point transformations: Preservation of an invariant torus under perturbations
A perturbed mapping is defined as a mapping of the form \(\Pi:(v,\varphi)\to(Av+f_0(\varphi)+f(v,\varphi),\, \varphi+g_0(\varphi)+g(v,\varphi))\), where \(A: V\to V\) is a linear bounded \(\|A\|<1\) operator, \(f_0(\varphi)\) and \(g_0(\varphi)\) are \(2\pi\)-periodic functions of the vector argument \(\varphi\in \mathbb{R}^p\) of the class \(C^1\) ranging in \(V\) and \(\mathbb{R}^p\) respectively, the equation \(\widetilde{\varphi}=\varphi+g_0(\varphi)\) has a unique solution \(\varphi=S_0(\tilde{\varphi})\in\mathbb{R}^p\) and \(\text{det}(I+g_0'(\varphi))\neq 0\) for all \(\varphi\in\mathbb{R}^p\). The perturbation \((f(v,\varphi), g(v,\varphi))\) belongs to the class \(\Sigma(\mu)\) of vector functions jointly continuous in \((v,\varphi)\) and \(2\pi\)-periodic with respect to the vector argument \(\varphi\), \(f(0,\varphi)\equiv 0\), \(g(0,\varphi)\equiv 0\) and \(\|f(v_1,\varphi_1)-f(v_2,\varphi_2)\|\leq\mu(\|v_1-v_2\|+\|\varphi_1-\varphi_2\|)\), \(\|g(v_1,\varphi_1)-g(v_2,\varphi_2)\|\leq\mu(\|v_1-v_2\|+\|\varphi_1-\varphi_2\|)\), for arbitrary \((v_k,\varphi_k)\), \(k=1,2\). Under some additional conditions, the authors prove that the perturbed mapping \(\Pi\) has an invariant torus close to the torus of the unperturbed mapping if the parameter \(\mu\) satisfying the perturbation \((f,g)\) is sufficiently small.
scientific article; zbMATH DE number 2176565