Bifurcations and Catastrophes Geometry of Solutions to Nonlinear Problems [Paperback]

Based on a lecture course, this text gives a rigorous introduction to nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach allowing a clear focus on the essential mathematical structures. It brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.

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 TrlÓÝ