Dynamical Systems VIII

Dieses Buch behandelt Anwendungen der Singularitätentheorie in der Mathematik. Die Beschreibung von Bifurkationsmengen, die in der Theorie der dynamischen Systeme und der Theorie der partiellen Differentialgleichungen auftreten, steht im Vordergrund. Alle vier Autoren sind bekannte Forscher auf dem Gebiet der Singularitätentheorie. Sie stellen zahlreiche neue Forschungsergebnisse vor und geben ungelöste Probleme und Vermutungen an. Das Buch wendet sich an Forscher und Studenten höherer Semester in der Mathematik und Physik, die Singularitäten verwenden.

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In the first volume of this survey (Arnol'd et al. (1988), hereafter cited as "EMS 6") we acquainted the reader with the basic concepts and methods of the theory of singularities of smooth mappings and functions. This theory has numerous applications in mathematics and physics; here we begin describing these applica tions. Nevertheless the present volume is essentially independent of the first one: all of the concepts of singularity theory that we use are introduced in the course of the presentation, and references to EMS 6 are confined to the citation of technical results. Although our main goal is the presentation of analready formulated theory, the readerwill also come upon some comparatively recent results, apparently unknown even to specialists. We pointout some of these results. 2 3 In the consideration of mappings from C into C in
3. 6 of Chapter 1, we define the bifurcation diagram of such a mapping, formulate a K(n, 1)-theorem for the complements to the bifurcation diagrams of simple singularities, give the definition of the Mond invariant N in the spirit of "hunting for invariants", and we draw the reader's attention to a method of constructing the image of a mapping from the corresponding function on a manifold with boundary. In
4. 6 of the same chapter we introduce the concept of a versal deformation of a function with a nonisolated singularity in the dass of functions whose critical sets are arbitrary complete intersections of fixed dimension.