Qualitative analysis of Newton's flow

The Newton flow of a parameter-dependent mapping is analysed in a neighbourhood of a singular point. It is shown that the Liapunov-Schmidt reduction process yields an invariant, exponentially attracting manifold of the Newton flow. If the singular point is a fold then a normal form of the reduced 2-D Newton flow is given. Moreover, numerical experiments suggest that discrete versions of the normal form seem to be useful for an understanding of the performance of the ordinary Newton method in a neighbourhood of the fold.