Necessary and sufficient conditions for the geometrical regularization of singularities

Abstract

We consider singularities of ordinary differential equations similar to the ones which appear in Celestial Mechanics. Using blow-up and time scaling the singular point is replaced by a manifold. In this manifold the flow is smooth, gradient-1ike and the critical points are (generically) hyperbolic. We look for conditions which assure regularizability of the singularity. Four conditions concerning the invariant manifolds of the critical points on the manifold, the variational equations along these manifolds and the eigenvalues of the linear part of the field at the critical points are necessary. It is proved that they are also sufficient. Several exemples are included.