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Smooth Invariant Manifolds and Normal Forms : World Scientific Series on Nonlinear Science Series A - A. Ya Kopanskii

Smooth Invariant Manifolds and Normal Forms

World Scientific Series on Nonlinear Science Series A

Hardcover Published: 1994
ISBN: 9789810215729
Number Of Pages: 396

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This volume deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and periodic trajectory. A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.

Introduction
Topological properties of flows and cascades in the vicinity of a rest point and a periodic trajectory
Basic definitions and factsp. 1
Hyperbolic rest pointsp. 8
Floquet-Lyapunov theoryp. 13
Hadamard-Bohl-Perron theoryp. 21
Finitely smooth normal forms of vector fields and diffeomorphisms
The problem on reducing a C[superscript[infinity]] vector field to C[superscript k] normal form in the vicinity of a hyperbolic rest pointp. 29
Normalization of jets of vector fields and diffeomorphismsp. 34
Polynomial normal formsp. 47
Simplification of the resonant normal form via finitely smooth transformationsp. 67
A general condition for C[superscript k] linearizabilityp. 76
C[superscript k] linearization theoremsp. 89
Some sufficient conditions for C[superscript k] linearizabilityp. 108
Theorems on C[superscript k] normal formsp. 136
Linearization of finitely smooth vector fields and diffeomorphismsp. 143
Normal forms (a supplement to [section] 8)p. 178
Summary of results on finitely smooth normal formsp. 182
Linear extensions of dynamical systems
Basic notions and factsp. 193
Exponential separation and exponential splittingp. 200
The structure of linear extensionsp. 206
Quadratic Lyapunov functionsp. 224
Weak regularity and Green-Samoilenko functionsp. 230
Smooth linear extensionsp. 251
Invariant subbundles of weakly non-linear extensions
Invariant subbundles and their intrinsic characterizationp. 261
The decomposition theoremp. 268
The Grobman-Hartman theoremp. 274
Smooth invariant subbundlesp. 278
Invariant manifolds
Persistence of invariant manifoldsp. 290
Normal hyperbolicity and persistencep. 293
Necessary condition for persistencep. 300
Asymptotic phasep. 304
Normal forms in the vicinity of an invariant manifold
Polynomial normal forms (the nodal case)p. 318
Polynomial normal forms (the saddle case)p. 332
Appendix. Some facts from global analysisp. 343
Bibliographyp. 364
Subject indexp. 380
List of symbolsp. 384
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9789810215729
ISBN-10: 981021572X
Series: World Scientific Series on Nonlinear Science Series A : Book 7
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 396
Published: 1994
Country of Publication: SG
Dimensions (cm): 24.77 x 16.51  x 2.54
Weight (kg): 0.84

Earn 435 Qantas Points
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