Implicit differential equations near a singular point

Abstract

This paper presents general and precise results on existence and number of solutions to implicitly defined ordinary differential equations in the vicinity of a singular point, where the equation is not equivalent to an explicit one. Unlike in most, if not in all, other studies devoted to this problem in the literature, it is not assumed that either rank drop occurs on an entire neighborhood of the singular point or that the equation is a scalar one. Attention is confined to the simplest but most frequently encountered kind of singular points. It is shown that such “standard” singular points split into two complementary classes: those from which exactly two distinct solutions emanate and those at which exactly two distinct solutions terminate. Whether a point of the latter class is eventually encountered is unaffected by slight modifications of the (nonsingular) initial condition and further evolution of the system governed by the singular ODE then requires using a suitable jump condition according to appropriate physical criteria. Points of the former class are also shown to play a more subtle but equally important role in the dynamics. The phenomena described here are relevant in various problems from the sciences, such as phase transitions or plasticity. They should also be relevant in some aspects of the classical domain of application of differential — algebraic equations: singular perturbations of ODE's.