Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

This paper addresses stability properties of singular equilibria arising in quasilinear implicit ODEs. Under certain assumptions, local dynamics near a singular point may be described through a continuous or directionally continuous vector field. This fact motivates a classification of geometric singularities into weak and strong ones. Stability in the weak case is analyzed through certain linear matrix equations, a singular version of the Lyapunov equation being especially relevant in the study. Weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points. Regarding strong equilibria, stability is proved via a Lyapunov–Schmidt approach under additional hypotheses. The results are shown to be relevant in singular root-finding problems.