Elements of the Mathematical Theory of Multi-Frequency Oscillations [electronic resource] /

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 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.

Bibliographic Details

Main Authors:	Samoilenko, A. M. author., SpringerLink (Online service)
Format:	Texto biblioteca
Language:	eng
Published:	Dordrecht : Springer Netherlands : Imprint: Springer, 1991
Subjects:	Mathematics., Mathematical analysis., Analysis (Mathematics)., Analysis.,
Online Access:	http://dx.doi.org/10.1007/978-94-011-3520-7
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