Catastrophe Theory

Second Edition

Domencio Castrigiano, Sandra Hayes

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9780429981432
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Catastrophe Theory was introduced in the 1960s by the renowned Fields Medal mathematician René Thom as a part of the general theory of local singularities. Since then it has found applications across many areas, including biology, economics, and chemical kinetics. By investigating the phenomena of bifurcation and chaos, Catastrophe Theory proved to

Chapter 1

Nondegenerate Critical Points: The Morse Lemma

The most basic notion of Catastrophe Theory is introduced in this chapter, namely, that of a critical point of a smooth function. This notion is also referred to in the standard literature as a singular point or a singularity. An essential distinction is made by dividing these points into two classes—the non-degenerate and the degenerate critical points. The degenerate ones are more difficult to handle and will be discussed in later chapters, whereas the nondegenerate critical points can be completely treated here.

The main theorem about nondegenerate critical points is Morse’s Lemma, which classifies all such points. Roughly speaking, the Morse Lemma states that locally around a nondegenerate critical point a smooth function of n real variables x₁, …, x_n can be transformed to a simple standard form by changing coordinates. There are exactly n + 1 such forms, and these are the quadratic forms

−x12−⋯−xs2+xs+12+⋯+xn2,

where s = 0, l, …, n. To each function corresponds exactly one canonical quadratic form.

The proof of Morse’s Lemma given here is based on the Taylor expansion of a smooth function, on the classification of quadratic forms on Rⁿ, and on a lemma locally diagonalizing a symmetric matrix of smooth functions. This lemma also plays a central role in proving the Reduction Lemma in Chapter 3. The classification of quadratic forms follows immediately from Sylvester’s Law of Inertia, which will also be proved here for the sake of completeness.

The treatment of nondegenerate critical points in this chapter is an indispensable first step...

Table of contents

• Cover
• Half Title
• Title page
• Copyright Page
• Dedication
• Table of Contents
• Foreword
• Preface to the First Edition
• Preface to the Second Edition
• 1 Nondegenerate Critical Points: The Morse Lemma
• 2 The Fold and the Cusp
• 3 Degenerate Critical Points: The Reduction Lemma
• 4 Determinacy
• 5 Codimension
• 6 The Classification Theorem for Germs of Codimension at Most 4
• 7 Unfoldings
• 8 Transversality
• 9 The Malgrange–Mather Preparation Theorem
• 10 The Fundamental Theorem on Universal Unfoldings
• 11 Genericity
• 12 Stability
• Appendix
• References
• Notation Index
• Subject Index