Bifurcations and Catastrophes

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.

Both author and translator are very well-known specialists Book is based on course at the prestigious Ecole Polytechnique, Paris Includes supplementary material: sn.pub/extras

Zusammenfassung

"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 Poincaré 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)

Inhalt

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 OrbitsStructural 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.