Modulus of orbital analytic classification for a family unfolding a saddle-node

In this paper we consider generic families of 2-dimensional analytic vector fields unfolding a generic (codimension 1) saddle-node at the origin. We show that a complete modulus of orbital analytic classification for the family is given by an unfolding of the Martinet–Ramis modulus of the saddle-node. The Martinet–Ramis modulus is given by a pair of germs of diffeomorphisms, one of which is an affine map. We show that the unfolding of this diffeomorphism in the modulus of the family is again an affine map. The point of view taken is to compare the family with the “model family” $(x^2-\varepsilon)\frac{\partial}{\partial x}+y(1+a(\varepsilon)x)\frac{\partial}{\partial y}$. The nontriviality of the Martinet–Ramis modulus implies geometric “pathologies” for the perturbed vector fields, in the sense that the deformed family does not behave as the standard family.