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Bifurcations and Catastrophes : Geometry of Solutions to Nonlinear Problems - D. Chillingworth

Bifurcations and Catastrophes

Geometry of Solutions to Nonlinear Problems

By: Michel Demazure, D. Chillingworth (Translator)

Paperback

Published: 15th December 1999
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Based on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.

"This book gives an introduction to the basic ideas in dynamical systems and catastrophe and bifurcation theory. It starts with the geometrical concepts which are necessary for the rest of the book. In the first four chapters, the author introduces the notion of local inversion for maps, submanifolds, tranversality, and the classical theorems related to the local theory of critical points, that is, Sard's theorem and Morse's lemma. After a study of the classification of differentiable maps, he introduces the notion of germ and shows how catastrophe theory can be used to classify singularities; elementary catastrophes are discussed in Chapter 5. Vector fields are the subject of the rest of the book. Chapter 6 is devoted to enunciating the existence and uniqueness theorems for ordinary differential equations; the notions of first integral, one-parameter group and phase portrait are also introduced in this part. Linear vector fields and the topological classification of flows are studied in Chapter 7. Chapter 8 is devoted to the classification of singular points of vector fields. Lyapunov theory and the theorems of Grobman and Hartman are also described in this chapter. The notions of Poincar(c) map and closed orbit, and the concepts necessary for the classification of closed orbits, are the principal ideas of Chapter 9; this chapter finishes with the notion of structural stability and the classification of structurally stable vector fields in dimension 2 and Morse-Smale vector fields. Finally, in Chapter 10 the author defines the idea of bifurcation of phase portraits and describes the simplest local bifurcations: saddle-node bifurcation, Hopf bifurcation, etc.

This book can be used as a textbook for a first course on dynamical systems and bifurcation theory." (Joan Torregrosa, Mathematical Reviews)

Introduction
Notation
Local Inversion
Introduction
A Preliminary Statement
Partial Derivatives. Strictly Differentiable Functions
The Local Inversion Theorem: General Statement
Functions of Class Cr
The Local Inversion Theorem for Cr maps
Generalizations of the Local Inversion Theorem
Submanifolds
Introduction
Definitions of Submanifolds
First Examples
Tangent Spaces of a Submanifold
Transversality: Intersections
Transversality: Inverse Images
The Implicit Function Theorem
Diffeomorphisms of Submanifolds
Parametrizations, Immersions and Embeddings
Proper Maps: Proper Embeddings
From Submanifolds to Manifolds
Some History
Transversality Theorems
Introduction
Countability Properties in Topology
Negligible Subsets
The Complement of the Image of a Submanifold
Sard''s Theorem
Critical Points, Submersions and the Geometrical Form of Sard''s Theorem
The Transversality Theorem: Weak Form
Jet Spaces
The Thom Transversality Theorem
Some History
Classification of Differentiable Functions
Introduction
Taylor Formulae Without Remainder
The Problem of Classification of Maps
Critical Points: the Hessian Form
The Morse Lemma
Fiburcations of Critical Points
Apparent Contour of a Surface in R3
Maps from R2 into R2
Envelopes of Plane Curves
Caustics
Genericity and Stability
Catastrophe Theory
Introduction
The Language of Germs
r-sufficient Jets; r-determined Germs
The Jacobian Ideal
The Theorem on Sufficiency of Jets
Deformations of a Singularity
The Principles of Catastrophe Theory
Catastrophes of Cusp Type
A Cusp Example
Liquid-Vapour Equilibrium
The Elementary Catastrophes
Catastrophes and Controversies
Vector Fields
Introduction
Exemples of Vector Fields (Rn Case)
First Integrals
Vector Fields on Submanifolds
The Uniqueness Theorem and Maximal Integral Curves
Vector Fields on Submanifolds
One-parameter Groups of Diffeomorphisms
The Existence Theorem (Local Case)
The Existence Theorem (Global Case)
The Integral Flow of a Vector Field
The Main Features of a Phase Portrait
Discrete Flows and Continuous Flows
Linear Vector Fields
Introduction
The Spectrum of an Endomorphism
Space Decomposition Corresponding to Partition of the Spectrum
Norm and Eigenvalues
Contracting, Expanding and Hyperbolic Endommorphisms
The Exponential of an Endomorphism
One-parameter Groups of Linear Transformations
The Image of the Exponential
Contracting, Expanding and Hyperbolic Exponential Flows
Topological Classification of Linear Vector Fields
Topological Classification of Automorphisms
Classification of Linear Flows in Dimension 2
Singular Pints of Vector Fields
Introduction
The Classification Problem
Linearization of a Vector Field in the Neighbourhodd of a Singular Point
Difficulties with Linearization
Singularities with Attracting Linearization
Liapunov Theory
The Theorems of Grobman and Hartman
Stable and Unstable Manifolds of a Hyperbolic Singularity
Differentiable Linearization: Statement of the Problem
Differentiable Linearization: Resonances
Differentiable Linearization: The Theorems of Sternberg and Hartman
Linearization in Dimenension 2
Some Historical Landmarks
Closed Orbits - Structural Stability
Introduction
The Poincarè Map
Characteristic Multipliers of a Closed Orbit
Attracting Closed Orbits
Classification of Closed Orbits and Classification of Diffeomorphisms
Hyperbolic Closed Orbits
Local Structural Stability
The Kupka-Smale Theorem
Morse-Smale Fields
Structural Stability Through the Ages
Bifurcations of Phase Portrait
Introduction
What Do We Mean by a Bifurcation?
The Centre Manifold Theorem
The Saddle-Node Bifurcation
The Hopf Bifurcation
Local Bifurcations Carried by a Closed Orbit
Saddle
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540521181
ISBN-10: 3540521186
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 304
Published: 15th December 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.57 x 15.65  x 1.68
Weight (kg): 0.41