Elements of the Mathematical Theory of Multi-Frequency Oscillations

Elements of the Mathematical Theory of Multi-Frequency Oscillations

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Description

Translated from the original Russian edition of 1987 (Nauka, Moscow), this volume deals with the theory of multi-frequency oscillations as a motion of a dynamical system which describes a recurrent trajectory on an invariant toroidal manifold of the system. In this way, the invariant toroidal manifo
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Product details

  • Hardback | 313 pages
  • 155 x 235 x 19.05mm | 1,430g
  • Dordrecht, Netherlands
  • English
  • 1991 ed.
  • XVI, 313 p.
  • 0792314387
  • 9780792314387

Table of contents

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 $$
C\left( {\mathcal{T}_m } \right)
$$.- 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 nonlinear 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.
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