On Pleated Singular Points of First-Order Implicit Differential Equations

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. 1 This approach goes back to H. Poincaré (Mémoire sur les courbes définies par leséquations différentielles. -J. Math. Pures Appl., Sér 4, 1885) and A. Clebsch (Ueber eine Fundamentalaufgabe der Invariantentheorie. -Göttingen Abh. XVII, 1872). The latter paper contains geometric interpretation of differential equations (both ordinary and partial) and related theory of connexes, which is quite similar to the lifting; an account of these ideas is contained also in the famous book "Vorlesungenüber höhere Geometrie" by F. Klein.

Asymptotic lines and parabolic points of plane fields in $\mathbb{R}^3$

“…For related works about local and global properties of implicit differential equations see for example [8], [10], [11], [18] and [37]. For global aspects of extrinsic geometry of non integrable plane fields in dimension three see [2], [5] and [24].…”