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Bifurcations and Catastrophes Geometry of Solutions to Nonlinear Problems

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 wh...

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Bibliographic Details
Main Author: Demazure, Michel
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 2000, 2000
Edition:1st ed. 2000
Series:Universitext
Subjects:
Geometry, Differential
Dynamical Systems
Manifolds (mathematics)
Differential Geometry
Global Analysis (mathematics)
Global Analysis And Analysis On Manifolds
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 7.9 Contracting, Expanding and Hyperbolic Exponential Flows
  • 7.10 Topological Classification of Linear Vector Fields
  • 7.11 Topological Classification of Automorphisms
  • 7.12 Classification of Linear Flows in Dimension 2
  • 8. Singular Points of Vector Fields
  • 8.1 Introduction
  • 8.2 The Classification Problem
  • 8.3 Linearization of a Vector Field in the Neighbourhood of a Singular Point
  • 8.4 Difficulties with Linearization
  • 8.5 Singularities with Attracting Linearization
  • 8.6 Lyapunov Theory
  • 8.7 The Theorems of Grobman and Hartman
  • 8.8 Stable and Unstable Manifolds of a Hyperbolic Singularity
  • 8.9 Differentable Linearization: Statement of the Problem
  • 8.10 Differentiable Linearization: Resonances
  • 8.11 DifferentiableLinearization: the Theorems of Sternberg and Hartman
  • 8.12 Linearization in Dimension 2
  • 8.13 Some Historical Landmarks
  • 9. Closed Orbits—Structural Stability
  • 9.1 Introduction
  • 9.2 The Poincaré Map
  • 1. Local Inversion
  • 1.1 Introduction
  • 1.2 A Preliminary Statement
  • 1.3 Partial Derivatives. Strictly Differentiable Functions
  • 1.4 The Local Inversion Theorem: General Statement
  • 1.5 Functions of Class Cr
  • 1.6 The Local Inversion Theorem for Crmaps
  • 1.7 Curvilinear Coordinates
  • 1.8 Generalizations of the Local Inversion Theorem
  • 2. Submanifolds
  • 2.1 Introduction
  • 2.2 Definitions of Submanifolds
  • 2.3 First Examples
  • 2.4 Tangent Spaces of a Submanifold
  • 2.5 Transversality: Intersections
  • 2.6 Transversality: Inverse Images
  • 2.7 The Implicit Function Theorem
  • 2.8 Diffeomorphisms of Submanifolds
  • 2.9 Parametrizations, Immersions and Embeddings
  • 2.10 Proper Maps; Proper Embeddings
  • 2.11 From Submanifolds to Manifolds
  • 2.12 Some History
  • 3. Transversality Theorems
  • 3.1 Introduction
  • 3.2 Countability Properties in Topology
  • 3.3 Negligible Subsets
  • 3.4 The Complement of the Image of a Submanifold
  • 3.5 Sard’s Theorem
  • 9.3 Characteristic Multipliers of a Closed Orbit
  • 9.4 Attracting Closed Orbits
  • 9.5 Classification of Closed Orbits and Classification of Diffeomorphisms
  • 9.6 Hyperbolic Closed Orbits
  • 9.7 Local Structural Stability
  • 9.8 The Kupka-Smale Theorem
  • 9.9 Morse-Smale Fields
  • 9.10 Structural Stability Through the Ages
  • 10.Bifurcations of Phase Portraits
  • 10.1 Introduction
  • 10.2 What Do We Mean by Bifurcation?
  • 10.3 The Centre Manifold Theorem
  • 10.4 The Saddle-Node Bifurcation
  • 10.5 The Hopf Bifurcation
  • 10.6 Local Bifurcations of a Closed Orbit
  • 10.7 Saddle-node Bifurcation for a Closed Orbit
  • 10.8 Period-doubling Bifurcation
  • 10.9 Hopf Bifurcation for a Closed Orbit
  • 10.10 An Example of a Codimension 2 Bifurcation
  • 10.11 An Example of Non-local Bifurcation
  • References
  • Notation
  • 5.11 The Elementary Catastrophes
  • 5.12 Catastrophes and Controversies
  • 6. Vector Fields
  • 6.1 Introduction
  • 6.2 Examples of Vector Fields (Rn Case)
  • 6.3 First Integrals
  • 6.4 Vector Fields on Submanifolds
  • 6.5 The Uniqueness Theorem and Maximal Integral Curves
  • 6.6 Vector Fields and Line Fields. Elimination of the Time
  • 6.7 One-parameter Groups of Diffeomorphisms
  • 6.8 The Existence Theorem (Local Case)
  • 6.9 The Existence Theorem (Global Case)
  • 6.10 The Integral Flow of a Vector Field
  • 6.11 The Main Features of a Phase Portrait
  • 6.12 Discrete Flows and Continuous Flows
  • 7. Linear Vector Fields
  • 7.1 Introduction
  • 7.2 The Spectrum of an Endomorphism
  • 7.3 Space Decomposition Corresponding to Partition of the Spectrum
  • 7.4 Norm and Eigenvalues
  • 7.5 Contracting, Expanding and Hyperbolic Endomorphisms
  • 7.6 The Exponential of an Endomorphism
  • 7.7 One-parameter Groups of Linear Transformations
  • 7.8 The Image of the Exponential
  • 3.6 Critical Points, Submersions and the Geometrical Form of Sard’s Theorem
  • 3.7 The Transversality Theorem: Weak Form
  • 3.8 Jet Spaces
  • 3.9 The Thorn Transversality Theorem
  • 3.10 Some History
  • 4. Classification of Differentiable Functions
  • 4.1 Introduction
  • 4.2 Taylor Formulae Without Remainder
  • 4.3 The Problem of Classification of Maps
  • 4.4 Critical Points: the Hessian Form
  • 4.5 The Morse Lemma
  • 4.6 Bifurcations of Critical Points
  • 4.7 Apparent Contour of a Surface in R3
  • 4.8 Maps from R2 into R2
  • 4.9 Envelopes of Plane Curves
  • 4.10 Caustics
  • 4.11 Genericity and Stability
  • 5. Catastrophe Theory
  • 5.1 Introduction
  • 5.2 The Language of Germs
  • 5.3 r-sufficient Jets; r-determined Germs
  • 5.4 The Jacobian Ideal
  • 5.5 The Theorem on Sufficiency of Jets
  • 5.6 Deformations of a Singularity
  • 5.7 The Principles ofCatastrophe Theory
  • 5.8 Catastrophes of Cusp Type
  • 5.9 A Cusp Example
  • 5.10 Liquid-Vapour Equilibrium

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