Analytical Moduli for Unfoldings of Saddle-Node Vector Fields

In this paper we consider germs of $k$-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension $k$ and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the germ of a vector field with a saddle-node. The point of view is to compare the family with a “model family” via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences for solutions of the cohomological equation.