On pleated singular points of first order implicit differential equations

TL;DR

This paper classifies the phase portraits of first order implicit differential equations near pleated singular points, revealing only six fundamentally different local behaviors despite the lack of a comprehensive classification.

Contribution

It provides a topological classification of phase portraits around pleated singular points, identifying six distinct types despite the absence of a full local classification.

Findings

Six essentially different phase portraits identified

Classification holds even without a local explicit classification

Provides a topological understanding of singular points

Abstract

We study phase portraits of a first order implicit differential equation in a neighborhood of its pleated singular point that is a non-degenerate singular point of the lifted field. Although there is no a visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there exist only six essentially different phase portraits, which are presented.