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.
IntroductionNotation1. Local Inversion1.1 Introduction1.2 A Preliminary Statement1.3 Partial Derivatives. Strictly Differentiable Functions1.4 The Local Inversion Theorem: General Statement1.5 Functions of Class Cr1.6 The Local Inversion Theorem for Cr maps1.8 Generalizations of the Local Inversion Theorem2. Submanifolds2.1 Introduction2.2 Definitions of Submanifolds2.3 First Examples2.4 Tangent Spaces of a Submanifold2.5 Transversality: Intersections2.6 Transversality: Inverse Images2.7 The Implicit Function Theorem2.8 Diffeomorphisms of Submanifolds2.9 Parametrizations, Immersions and Embeddings2.10 Proper Maps: Proper Embeddings2.11 From Submanifolds to Manifolds2.12 Some History3. Transversality Theorems3.1 Introduction3.2 Countability Properties in Topology3.3 Negligible Subsets3.4 The Complement of the Image of a Submanifold3.5 Sard's Theorem3.6 Critical Points, Submersions and the Geometrical Form of Sard's Theorem3.7 The Transversality Theorem: Weak Form3.8 Jet Spaces3.9 The Thom Transversality Theorem3.10 Some History4. Classification of Differentiable Functions4.1 Introduction4.2 Taylor Formulae Without Remainder4.3 The Problem of Classification of Maps4.4 Critical Points: the Hessian Form4.5 The Morse Lemma4.6 Fiburcations of Critical Points4.7 Apparent Contour of a Surface in R34.8 Maps from R2 into R2.4.9 Envelopes of Plane Curves4.10 Caustics4.11 Genericity and Stability5. Catastrophe Theory5.1 Introduction5.2 The Language of Germs5.3 r-sufficient Jets; r-determined Germs5.4 The Jacobian Ideal5.5 The Theorem on Sufficiency of Jets5.6 Deformations of a Singularity5.7 The Principles of Catastrophe Theory5.8 Catastrophes of Cusp Type5.9 A Cusp Example5.10 Liquid-Vapour Equilibrium5.11 The Elementary Catastrophes5.12 Catastrophes and Controversies6. Vector Fields6.1 Introduction6.2 Exemples of Vector Fields (Rn Case)6.3 First Integrals6.4 Vector Fields on Submanifolds6.5 The Uniqueness Theorem and Maximal Integral Curves6.6 Vector Fields on Submanifolds6.7 One-parameter Groups of Diffeomorphisms6.8 The Existence Theorem (Local Case)6.9 The Existence Theorem (Global Case)6.10 The Integral Flow of a Vector Field6.11 The Main Features of a Phase Portrait6.12 Discrete Flows and Continuous Flows7. Linear Vector Fields7.1 Introduction7.2 The Spectrum of an Endomorphism7.3 Space Decomposition Corresponding to Partition of the Spectrum7.4 Norm and Eigenvalues7.5 Contracting, Expanding and Hyperbolic Endommorphisms7.6 The Exponential of an Endomorphism7.7 One-parameter Groups of Linear Transformations7.8 The Image of the Exponential7.9 Contracting, Expanding and Hyperbolic Exponential Flows7.10 Topological Classification of Linear Vector Fields7.11 Topological Classification of Automorphisms7.12 Classification of Linear Flows in Dimension 28 Singular Pints of Vector Fields8.1 Introduction8.2 The Classification Problem8.3 Linearization of a Vector Field in the Neighbourhodd of a Singular Point8.4 Difficulties with Linearization8.5 Singularities with Attracting Linearization8.6 Liapunov Theory8.7 The Theorems of Grobman and Hartman8.8 Stable and Unstable Manifolds of a Hyperbolic Singularity8.9 Differentiable Linearization: Statement of the Problem8.10 Differentiable Linearization: Resonances8.11 Differentiable Linearization: The Theorems of Sternberg and Hartman8.12 Linearization in Dimenension 28.13 Some Historical Landmarks9 Closed Orbits - Structural Stability9.1 Introduction9.2 The Poincaré Map9.3 Characteristic Multipliers of a Closed Orbit9.4 Attracting Closed Orbits9.5