Smooth normal forms of folded elementary singular points

Abstract

In this work the smooth and topological normal forms of the first-order implicit differential equations in the plane near its folded degenerate elementary singular point are found, and thereby, the smooth and topological classification of folded elementary singular points of these equations is completed. It is proved, for example, that the equation is equivalent near its folded singular point of saddlenode type to some equation (dy/dx+x 2+Ax 3)2=y, whereA is a real number, in some appropriate smooth coordinate system in the plane with the origin at this point. The numberA is the parameter of the normal form.