Singularities, Bifurcations and Catastrophes

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.

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.

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).