Annulus principle in the problem of the existence of an infinite-dimensional invariant torus

An annulus-like set of the form K=B^ is under consideration, where B is a closed ball in a Banach space V and T^ is the so-called standard infinite-dimensional torus, defined by T^=E/2^, where E is an infinite-dimensional Banach space and Z^ is an abstract integer lattice in E. The main result is as follows: for a certain class of smooth maps K K we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form A=\ (v, ) K: v=h () V, ^{\}, where h () is a continuous function of ^. We also answer a number of related questions. First, we consider the problem of the Cᵐ-smoothness of the manifold A for each positive integer m; second, we show that all trajectories of the map with initial conditions in K tend to A and admit an asymptotic phase; third, we extend our results to semiflows and then apply the theory developed to integral networks of nonlinear oscillators. Bibliography: 42 titles.