Numer. Math. 48, 417-427 (1986) Numerische Mathematik 9 Springer-Verlag 1986 How to Get around a Simple Quadratic Fold F. Brezzi 1., M. Cornalba 2, A. Di Carlo 3.* x Istituto di Analisi Numerica del C.N.R. and Dipartimento di Meccanica Strutturale, Universit~t di Pavia, 1-27100 Pavia, Italy 2 Dipartimento di Matematica, Universitb. di Pavia, 1-27100 Pavia, Italy 3 Dipartimento di Ingegneria Strutturale e Geotecnica, Universit/t di Roma "La Sapienza", Via Eudossiana, 18, 1-00184 Roma, Italy Summary. We analyse from a theoretical point of view a computational technique previously introduced [-2, 5] for tracing branches of solutions of nonlinear equations near simple quadratic folds. Subject Classifications: AMS(MOS): 65H10; CR: G.1.5. 1. Introduction The aim of this paper is to analyse - from a mathematical point of view - a technique, first introduced in [-2, 5], for the computation of branches of solutions in the vicinity of a simple quadratic fold. Such a technique can be summarized as follows. Assume that we are given a nonlinear problem of the form: f(u, 2) =0, (1.1) where F is a smooth mapping from R"• into R". We say that a point (u o, 20)elR" x R (which, from now on, we shall assume to be the origin (0,0)) is a simple critical point for (1.1) if L:=D,F ~ has rank n-1. (1.2) Let us denote by q~0 and ~b* the generators of the kernels of L and of its transpose L*, normalized so that ll~oll = II~ll = 1. We say that (0,0) is a simple fold for (1.1) if it is a simple critical point and in addition <DxF ~ ~b~> ~=0, (1.3) * Supported in part by an M.P.I. 40% grant ** Supported in part by a C.N.R. grant