Behavior of solutions of nonlinear systems of difference equations in a neighborhood of an invariant toroidal manifold (Q1262473)

Consider a system of difference equations of the type (*) \(\phi_{n+1}=\phi_ n+a(\phi_ n,h_ n),\quad h_{n+1}=[E+P(\phi_ n,h_ n)]h_ n+C(\phi_ n),\) where \(\phi =(\phi^ 1,\phi^ 2,...,\phi^ m),\quad h=(h^ 1,h^ 2,...,h^{\ell}),\) and a(\(\phi\),h), P(\(\phi\),h), C(\(\phi)\) are some differentiable \(2\pi\)- periodic in \(\phi\) functions. \textit{A. M. Samoilenko}, the second author, and \textit{N. A. Perestyuk} [Diff. Uravn. 9, No.10, 1904-1910 (1973; Zbl 0269.39001)] have shown that there exists a continuous toroidal manifold \({\mathfrak I}_ m:h=u(\phi)\) for the system (*). In the present paper the authors prove that under suitable conditions every solution of (*) with initial data in the vicinity of \({\mathfrak I}_ m\) is close to some solution lying on \({\mathfrak I}_ m\).