Smooth Invariant Manifolds And Normal Forms
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Smooth Invariant Manifolds And Normal Forms
Author: Alexander Kopanskii
language: en
Publisher: World Scientific
Release Date: 1994-12-22
This book 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 a periodic trajectory. A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.
Normal Forms and Unfoldings for Local Dynamical Systems
Author: James Murdock
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-04-10
The subject of local dynamical systems is concerned with the following two questions: 1. Given an n×n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of di?erential equations having a rest point at the origin with linear part Ax, that is, all systems of the form x ? = Ax+··· , n where x? R and the dots denote terms of quadratic and higher order. 2. Describethebehavior(neartheorigin)ofallsystemsclosetoasystem of the type just described. To answer these questions, the following steps are employed: 1. A normal form is obtained for the general system with linear part Ax. The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. 2. An unfolding of the normal form is obtained. This is intended to be the simplest form into which all systems close to the original s- tem can be transformed. It will contain parameters, called unfolding parameters, that are not present in the normal form found in step 1. vi Preface 3. The normal form, or its unfolding, is truncated at some degree k, and the behavior of the truncated system is studied.
Poisson Structures and Their Normal Forms
Author: Jean-Paul Dufour
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-01-17
The aim of this book is twofold. On the one hand, it gives a quick, self-contained introduction to Poisson geometry and related subjects. On the other hand, it presents a comprehensive treatment of the normal form problem in Poisson geometry. Even when it comes to classical results, the book gives new insights. It contains results obtained over the past 10 years which are not available in other books.