Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equivalence of map-germs and finally presents the path formulation of bifurcation theory. This formulation, developed partly by the author, is more general and more flexible than the original one dating from the 1980s. A series of appendices discuss standard background material, such as calculus of several variables, existence and uniqueness theorems for ODEs, and some basic material on rings and modules. Based on the author's own teaching experience, the book contains numerous examples and illustrations. The wealth of end-of-chapter problems develop and reinforce understanding of the key ideas and techniques: solutions to a selection are provided.
- Builds up the mathematical background for a new approach to the foundations of bifurcation theory
- The first two parts are of independent interest and can be the basis for an advanced undergraduate or graduate course
- Contains many colour figures and exercises, with some solutions included and a manual for selected others available for teachers
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Reviews & endorsements
'This beautiful book is in fact a course which can be viewed as addressed to undergraduate and graduate students, to junior and senior researchers, to the teaching staff (faculty), and to other people interested in the ﬁeld.' Vladimir Răsvan, European Mathematical Society
'This mostly self-contained and user-friendly textbook, aimed at advanced undergraduate level and above, provides a careful and accessible introduction to methods of singularity theory that underlie much of local bifurcation theory.' D. R. J. Chillingworth, MathSciNet
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Product details
- Date Published: July 2021
- format: Paperback
- isbn: 9781316606216
- length: 448 pages
- dimensions: 243 x 169 x 23 mm
- weight: 0.86kg
- availability: Available
Table of Contents
Preface
1. What's It All About?
Part I. Catastrophe Theory
2. Families of Functions
3. The Ring of Germs of Smooth Functions
4. Right Equivalence
5. Finite Determinacy
6. Classification of the Elementary Catastrophes
7. Unfoldings and Catastrophes
8. Singularities of Plane Curves
9. Even Functions
Part II. Singularity Theory
10. Families of Maps and Bifurcations
11. Contact Equivalence
12. Tangent Spaces
13. Classification for Contact Equivalence
14. Contact Equivalence and Unfoldings
15. Geometric Applications
16. Preparation Theorem
17. Left-Right Equivalence
Part III. Bifurcation Theory
18. Bifurcation Problems and Paths
19. Vector Fields Tangent to a Variety
20. Kv-equivalence
21. Classification of Paths
22. Loose Ends
23. Constrained Bifurcation Problems
Part IV. Appendices
A. Calculus of Several Variables
B. Local Geometry of Regular Maps
C. Differential Equations and Flows
D. Rings, Ideals and Modules
E. Solutions to Selected Problems.
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Author
James Montaldi, University of Manchester
James Montaldi is Reader in Mathematics at University of Manchester. He has worked both in theoretical aspects of singularity theory as well as applications to dynamical systems, and co-edited the books: Geometric Mechanics and Symmetry: The Peyresq Lectures (Cambridge, 2005), Peyresq Lectures in Nonlinear Systems (2000), and Singularity Theory and its Applications Part 1 (1991).
Contributors
James Montaldi