Elements of the Mathematical Theory of Multi-Frequency Oscillations

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Mathematics and its Applications 71

ISBN:	9401055572
ISBN 13:	9789401055574
Autor:	Samoilenko, Anatolii M
Verlag:	Springer Verlag GmbH
Umfang:	xvi, 313 S.
Erscheinungsdatum:	25.09.2012
Auflage:	1/2012
Produktform:	Kartoniert
Einband:	Kartoniert

Inhaltsangabe1. 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. Quasiperiodic 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 rframe 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 nonlinear system. 4.7. Galerkin’s method for the construction of an invariant torus of a nonlinear 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.

Artikelnummer: 5452038 Kategorie: Analysis