Singularities and Groups in Bifurcation Theory

1. Front Matter

   Pages i-xvii

   PDF

2. A Brief Introduction to the Central Ideas of the Theory

   • Martin Golubitsky, David G. Schaeffer

   Pages 1-50

3. The Recognition Problem

   • Martin Golubitsky, David G. Schaeffer

   Pages 51-116

4. Unfolding Theory

   • Martin Golubitsky, David G. Schaeffer

   Pages 117-181

5. Classification by Codimension

   • Martin Golubitsky, David G. Schaeffer

   Pages 182-212

6. An Example of Moduli

   • Martin Golubitsky, David G. Schaeffer

   Pages 213-242

7. Bifurcation with Z ₂-Symmetry

   • Martin Golubitsky, David G. Schaeffer

   Pages 243-288

8. The Liapunov—Schmidt Reduction

   • Martin Golubitsky, David G. Schaeffer

   Pages 289-336

9. The Hopf Bifurcation

   • Martin Golubitsky, David G. Schaeffer

   Pages 337-396

10. Two Degrees of Freedom Without Symmetry

    • Martin Golubitsky, David G. Schaeffer

    Pages 397-416

11. Two Degrees of Freedom with (Z ₂ ⊕ Z ₂)-Symmetry

    • Martin Golubitsky, David G. Schaeffer

    Pages 417-453

12. Back Matter

    Pages 455-466

    PDF

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob­ lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.

• Area
• Bifurcations
• Group theory
• Groups
• Volume

• Department of Mathematics, University of Houston, Houston, USA

  Martin Golubitsky

• Department of Mathematics, Duke University, Durham, USA

  David G. Schaeffer

Policies and ethics