Singularities and groups in bifurcation theory. Volume I
type: Item
instance of: scholarly article
title: Singularities and groups in bifurcation theory. Volume I
zbMATH Open document ID: 0607.35004
author: D. G. Schaeffer
author: Martin Golubitsky
published in: Applied Mathematical Sciences
publication date: 1985-01-01T00:00:00Z
review text: Here is the first sentence in the Preface: ''This book has been written in a frankly partisan style - we believe that singularity theory offers an extremely useful approach to bifurcation problems and we hope to convert the reader to this view.'' Later, the authors precise carefully their role: ''Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory...to bifurcation problems.'' On the other hand, the book will be followed by a companion where group theory and symmetry will play a more important role. Indeed, the aim is to provide a rather systematic approach to the above topics. The recognition problem, perturbations and unfoldings, and classification and codimension, are strongly emphasized. They are developed at length, but this is balanced by the attention paid to applications. This balance is, in our opinion, one of the best qualities of the book. A second one is its introductory character, together with the importance given to motivation and guiding ideas. First, let us give a survey of the contents. Chapter I is an introduction, where some interesting examples are considered, together with a first version of the Lyapunov-Schmidt reduction and the calculation of the derivatives of the reduced function. Chapters II-IV are the core of the book. Chapter II is devoted to the recognition problem, i.e. to characterize mappings equivalent to a given map. This leads to the splitting of derivatives into low-order, higher- order and intermediate-order terms, and to the study of the restricted tangent space. Unfolding theory is the main subject of Chapter III. This means to find, up to equivalence, all perturbations for a given bifurcation problem. The Universal Unfolding Theorem is stated and used, but its proof is deferred until Vo. II. Codimension is characterized as the number of parameters entering in a universal unfolding. The second part deals with nonpersistent bifurcation diagrams and the t…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 74M20
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D30
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 3981627
zbMATH Keywords: singularity theory
zbMATH Keywords: bifurcation
zbMATH Keywords: symmetry
zbMATH Keywords: recognition
zbMATH Keywords: perturbations
zbMATH Keywords: unfoldings
zbMATH Keywords: classification
zbMATH Keywords: codimension
zbMATH Keywords: Lyapunov-Schmidt reduction
zbMATH Keywords: hysteresis
zbMATH Keywords: moduli
zbMATH Keywords: Hopf Bifurcation
zbMATH Keywords: buckling
zbMATH Keywords: Path Formulation
zbMATH Keywords: Catastrophe Theory
zbMATH Keywords: Singularity
reviewed by: Jesús Hernández
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3981627