Singularities of vector fields on the plane

The main aim of the paper is to prove that in generic Cm n-parameter families of vector fields on 2-dimensional manifolds, all singularities with characteristic orbits are topologically determined by a finite jet. Moreover, for each n, we prove that we have only a finite number of topological equivalence classes of codimension n singularities with characteristic orbits, and for n < 4 we give an explicit classification. We also show that finitely determined singularities on the plane satisfy a Eojasiewicz inequality, and as a consequence of the method we obtain a Cm “curve selection lemma ” for dimension 2. More precise statements and a survey of the contents can be found in Section 4 of this chapter. I wish to thank F. Takens for introducing me to this field, and for his valuable help and advice during the preparation of this paper. I also want to thank the Instituto de Matematica Pura e Aplicada for its hospitality during the preparation of part of this work.