The value of f(x) at x0 is infinity or cannot be defined and the function is called singular at x0. The singularity of mathematical function is point at which the function's value can not be defined or in other words value approaches infinity. As we use the term, bifurcation theory is the study of equations with multiple solutions. First let us consider singularity in mathematics. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. In Volume II the emphasis will be more balanced. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. 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 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. 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. 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. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. In this preface we will discuss what we feel are the strengths of the singularity theory approach. 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.

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