On the local classification of smooth maps induced by Newton's method

The classification of germs of smooth maps f:Rn→Rn induced by the continuous Newton method -f′(x)x˙=f(x) is addressed in this paper. This classification problem is shown to rely on an equivalence notion located between right and contact equivalences, driving the classification problem from the setting of quasilinear ODEs to the singularity theory framework. One-dimensional problems and regular, n-dimensional cases are easily characterized, and normal forms for them are given. Folded zeros in Rn display a much richer behavior: an invariant and a preliminary normal form are derived for these cases.