Singularities, Bifurcations and Catastrophes

Singularities, Bifurcations and Catastrophes

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James Montaldi
Cambridge University Press, 6/24/2021
EAN 9781316606216, ISBN10: 131660621X

Paperback, 448 pages, 24.4 x 17 x 2.7 cm
Language: English
Originally published in English

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.

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.