Stability of discrete systems in a critical case (Q1385947)

We consider the autonomous system of difference equations \[ x_{n+1} =Ax_n+f(x_n), \quad n\in\mathbb{N} \tag{1} \] that describes discrete systems of automatic control. Here \(x_n\in \mathbb{R}^l\), \(A\) is a square matrix of size \(l\times l\), and \(f(x_n)\) is an analytic vector-valued function that can be represented in the form \[ f(x_n)= \sum^\infty_{| k |=2} f^{(k)} (x_n), \] where \(f^{(k)} (x_n)= \sum_{| j |=k} f_kx^k_n\) are forms of the \(k\)th order. We study the stability of motions defined by the system (1) in the critical case when the matrix \(A\) has \(m\) pairs \((2m=1)\) of complex conjugate eigenvalues \(\lambda_k=\exp (i\varphi_k)\), \(\overline\lambda_k= \exp (-i\varphi_k)\).