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. Gårding'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 $$C\left( {\mathcal{T}_m } \right)$$.- 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