Applications of the blowing-up construction and algebraic geometry to bifurcation problems

Abstract

A generalization of the Morse lemma to vector-valued functions is proved by a blowing-up argument. This is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem. Bifurcation in a one- or multi-parameter problem is guaranteed if the leading term is of even degree (it is often two) and satisfies a regularity condition. Applications are given to nonlinear eigenvalue problems and to the Hopf bifurcation.

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