Contact equivalence and bifurcation theory

Guimaraes, Luiz Carlos (1979) Contact equivalence and bifurcation theory. University of Southampton, Doctoral Thesis.

Abstract

This thesis takes up a suggestion of L. Nirenberg that the theory of singularities of maps could play a useful role in the study of the solutions of a non-linear equation. The concept of contact equivalence is put forward as providing a natural framework for this study. In order to put this concept in a form suitable for application in bifurcation problems, the theory of contact deformations is extended to a class of germsin Banach spaces. The proofs of the main results of the theory are extended to this class, whenever possible by resorting to a reduction procedure and making use of the known finite-dimensional results. Also with the aim of making known results in singularity theory available in this now setting, the relationship between contact and right equivalence for germs of functions is examined. An essential ingredient in the theory of deformations is the notion of versality. A necessary and sufficient -condition for a deformation with parameters in a Banach space to be vernal is established. Two alternative equivalence relations are also examined from the point of view of their usefulness in bifurcation theory, and the possibility of a theory of deformations for these is briefly looked at.

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