Elements of the Mathematical Theory of Multi-Frequency Oscillations

1. Periodic and quasi-periodic functions.- 1.1. The function spaces $$ C^r left( {mathcal{T}_m } right) $$ and $$ H^r left( {mathcal{T}_m } right) $$.- 1.2. Structure of the spaces $$ H^r left( {mathcal{T}_m } right) $$. Sobolev theorems.- 1.3. Main inequalities in $$ C^r left( omega right) $$.- 1.4. Quasi-periodic functions. The spaces $$ H^r left( omega right) $$.- 1.5. The spaces $$ H^r left( omega right) $$ and their structure.- 1.6. First integral of a quasi-periodic function.- 1.7. Spherical coordinates of a quasi-periodic vector function.- 1.8. The problem on a periodic basis in En.- 1.9. Logarithm of a matrix in $$C^l left( {mathcal{T}_m } right)$$. Sibuja's theorem.- 1.10. Garding's inequality.- 2. Invariant sets and their stability.- 2.1. Preliminary notions and results.- 2.2. One-sided invariant sets and their properties.- 2.3. Locally invariant sets. Reduction principle.- 2.4. Behaviour of an invariant set under small perturbations of the system.- 2.5. Quasi-periodic motions and their closure.- 2.6. Invariance equations of a smooth manifold and the trajectory flow on it.- 2.7. Local coordinates in a neighbourhood of a toroidal manifold. Stability of an invariant torus.- 2.8. Recurrent motions and multi-frequency oscillations.- 3. Some problems of the linear theory.- 3.1. Introductory remarks and definitions.- 3.2. Adjoint system of equations. Necessary conditions for the existence of an invariant torus.- 3.3. Necessary conditions for the existence of an invariant torus of a linear system with arbitrary non-homogeneity in $$ Cleft( {mathcal{T}_m } right) $$.- 3.4. The Green's function. Sufficient conditions for the existence of an invariant torus.- 3.5. Conditions for the existence of an exponentially stable invariant torus.- 3.6. Uniqueness conditions for the Green's function and the properties of this function.- 3.7. Separatrix manifolds. Decomposition of a linear system.- 3.8. Sufficient conditions for exponential dichotomy of an invariant torus.- 3.9. Necessary conditions for an invariant torus to be exponentially dichotomous.- 3.10. Conditions for the $$C'left( {mathcal{T}_m } right)$$-block decomposability of an exponentially dichotomous system.- 3.11. On triangulation and the relation between the $$C'left( {mathcal{T}_m } right)$$)-block decomposability of a linear system and the problem of the extendability of an r-frame to a periodic basis in En.- 3.12. On smoothness of an exponentially stable invariant torus.- 3.13. Smoothness properties of Green's functions, the invariant torus and the decomposing transformation of an exponentially dichotomous system.- 3.14. Galerkin's method for the construction of an invariant torus.- 3.15. Proof of the main inequalities for the substantiation of Galerkin's method.- 4. Perturbation theory of an invariant torus of a non--linear system.- 4.1. Introductory remarks. The linearization process.- 4.2. Main theorem.- 4.3. Exponential stability of an invariant torus and conditions for its preservation under small perturbations of the system.- 4.4. Theorem on exponential attraction of motions in a neighbourhood of an invariant torus of a system to its motions on the torus.- 4.5. Exponential dichotomy of invariant torus and conditions for its preservation under small perturbations of the system.- 4.6. An estimate of the smallness of a perturbation and the maximal smoothness of an invariant torus of a non-linear system.- 4.7. Galerkin's method for the construction of an invariant torus of a non-linear system of equations and its linear modification.- 4.8. Proof of Moser's lemma.- 4.9. Invariant tori of systems of differential equations with rapidly and slowly changing variables.- Author index.- Index of notation.