Elements of the Mathematical Theory of Multi-Frequency Oscillations

It can be seen from the title that we shall be dealing here with a mathemati- cal theory based on precise notions and definitions. The central object of this theory is an invariant toroidal manifold of a dynamical system considered in a Euclidean space En or in its product with an n-dimensional torus, that is, in the space En X T m . Taking into account the fact that a quasi-periodic solution of a dynamical system in En "sweeps out" an invariant torus of this sytem, the relation between the theory of invariant tori and the theory of multi-frequency oscillations becomes clear; the existence of such tori is a necessary condition for the existence of multi-frequency oscillations of quasi-periodic solutions of dynamical systems. By defining multi-frequency oscillations as a motion of a dynamical system, describing a recurrent trajectory on an invariant toroidal manifold of the system, we make the invariant toroidal manifold into the main subject of the mathematical theory of multi-frequency oscillations; the exis- tence of such a manifold is sufficient for multi-frequency oscillations of the system to exist.
The original part of the monograph - the last two chapters, is devoted to the following topics: the existence of invariant toroidal manifolds for lin- ear systems in En X T m , perturbation theory of such manifolds for non-linear systems, smoothness and stability properties of these manifolds, the behaviour of trajectories in a small neighbourhood, the study of separatrix manifolds in the case of exponential dichotomy of toroidal manifolds, linear system block decomposition under exponential dichotomy, and substantiation of Galerkin's procedure for finding toroidal manifolds.
The first two chapters of the book contain a theory of quasi-periodic func- tions, expounded on the basis of the theory of periodic functions of several variables, and some results from the theory of invariant sets and their stability. The material expounded in these chapters is closely related to the theory of multi-frequency oscillations and will be useful to specialists who are not well acquainted with the above-mentioned topics.