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

Buchner, M., J. E. Marsden and S. Schecter

J. Differential Equations, 48, (1983), 404-433

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.