Bifurcations and Catastrophes

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.- 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.- 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.- 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.- 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.