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 Π K→ K we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form A=\(v,φ)∈ K: v=h(φ)∈ V, φ∞\, where h(φ) is a continuous function of φ∞. 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 Π 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.