On Codimension 3 Singular Points of First-order Implicit Differential Equations

Abstract

We investigate local phase portraits of first-order implicit differential equations in a neighborhood of their singular points of codimension 3. Namely, we consider singular points where the lifted field of the equation on a surface is non-singular, but the projection of the surface to the phase plane has a singularity of one of three types: lips, beaks, swallowtail. We also consider generic bifurcations in one-parameter families of such equations.