Bifurcation in Degenerate Directions

Abstract

A typical task in the practical treatment of bifurcation problems is the determination of the set of zeros of a mapping G: ℝⁿ⁺¹→ℝⁿ near a given singular solution z_o of G(z)=o. The approach of DESCLOUX and RAPPAZ [1] unifies preceding constructive methods as in [4], [5]. Following [1], one determines the so-called “characteristic rays” by computing the zeros of the multilinear mapping associated with the first nonvanishing derivative at zero of the Lyapunov-Schmidt reduction. Characteristic rays are the candidates for tangents to solution curves passing through z_o. Each characteristic ray σ satisfying a certain nondegeneracy condition corresponds to a branch of solutions of G(z)=o passing through z_o and tangent to σ at z_o. If all characteristic rays are nondegenerate, a complete description of the zero set of G near z_o is possible. However, there are applications where the above mentioned procedure gives no information about the local structure of G^-1{o}.