Catastrophe Theory - 2nd Edition

✍ Scribed by Vladimir Igorevich Arnol'd, V. I. Arnol'd

• Publisher: Springer-Verlag Telos
• Year: 1986
• Tongue: English
• Leaves: 120
• Edition: 2nd rev. and expanded ed
• Category: Library

✦ Synopsis

This well-known booklet, now in its third, expanded edition, provides an informal survey of applications of singularity theory in a wide range of areas. Although the first few chapters touch briefly (and critically) on the Thom-Zeeman catastrophe theory, most of the book is concerned with more recent and less controversial aspects, covering such topics as: bifurcations and stability loss, wavefront propagation, the distribution of matter in the universe, optimization and control problems, visible contours, bypassing an obstacle, symplectic and contact geometry, complex singularities, and the surprising connections between singularities and widely disparate mathematical objects such as regular polyhedra and reflection groups. Readers familiar with the previous editions will find much that is new. Results have been brought up to date, and among the new or expanded topics discussed are delayed loss of stability, cascades of period doublings and triplings, shock waves, implicit differential equations and folded singularities, interior scattering, and more. Three new sections give an overview of the history of singularity theory and its applications from Leonardo da Vinci to modern times, a discussion of perestroika in terms of the theory of metamorphoses, and a list of 93 problems touching on most of the subject matter in the book. The text is enhanced by fifteen new drawings (there are now 87 in all) and improvements to old ones. The already extensive literature list has been updated and expanded. As a result, the book has been enlarged by almost a third. Arnol'd's goal with this edition remains the same: to explain the essence of the results and applications to readers having a minimal mathematical background. All that he asks, is that the reader have an inquiring mind.

✦ Table of Contents

Preface to the Second English Edition......Page 5
Preface to the First English Edition......Page 8
Preface to the First Russian Edition......Page 9
Contents......Page 11
1 Singularities, Bifurcations, and Catastrophes......Page 13
2 Whitney's Singularity Theory......Page 15
3 Applications of Whitney's Theory......Page 19
4 A Catastrophe Machine......Page 22
5 Bifurcations of Equilibrium States......Page 26
6 Loss o f Stability o f Equilibrium and o f Self-Oscillating
Modes of Behaviour......Page 32
7 Singularities of the Stability Boundary and the Principle
of the Fragility of Good Things......Page 39
8 Caustics, Wave Fronts, and Their Metamorphoses......Page 41
10 Singularities in Optimization Problems : the Maximum
Function......Page 56
11 Singularities of the Boundary of Attainability......Page 59
12 Smooth Surfaces and Their Projections......Page 69
13 The Problem of Bypassing an Obstacle......Page 75
14 Symplectic and Contact Geometry......Page 79
15 Complex Singularities......Page 88
16 The Mysticism of Catastrophe Theory......Page 101
Bibliographical Comments......Page 107