Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

Abstract

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. Academic Press

On Some Aspects of the Geometry of Differential Equations in Physics

“…The aim of the paper is to classify the normal forms for generalized vector fields. Later papers have studied normal forms [73] and stability [84] of such equations. Finally, a brief review on singular system of differential equations is presented in [33], where their geometric features, including a geometric method for obtaining a nonnumerical solution, are analized.…”

A geometric singular perturbation theory approach to constrained differential equations

“…Systems such as will be called constrained systems , see . Other possible names are quasilinear differential equations , generalized vector fields , or even differential‐algebraic equations (DAE), since some relations in may not include any of the derivatives of the components of the state vector , yielding a set with some algebraic equations together with some differential equations.…”

“…Systems of the form differ from autonomous ordinary differential equations due to the existence of the so‐called impasse set which is given by The main reason for this name is that a solution of can reach an impasse point at some time t 0 , and then could not be defined anymore when . It is also common to use the word singular to refer to the points in , since at these points the matrix is singular (see, e.g., ).…”

Equilibrios singulares en ecuaciones diferenciales algebraicas: aplicación a redes eléctricas no lineales

“…En este capítulo trabajaremos con EDAs cuasilineales singulares deíndice 0 y enunciaremos resultados conocidos para problemas estacionarios y no estacionarios. Ilustraremos el problema estacionario con un ejemplo sobre el método continuo de Newton [17], [18], [19].…”

“…Porúltimo, utilizamos el algoritmo de desingularización para establecer una clasificación sobre los distintos tipos v vi de equilibrio en estos modelos.En el Capítulo 3 se establece la clasificación de las EDAs cuasilineales singulares en estacionarias y no estacionarias. Para ilustrar la clase de EDAs cuasilineales singulares estacionarias, se presenta un ejemplo de aplicación en el método continuo de Newton[19] [17]. Para el caso de las EDAs cuasilineales singulares no estacionarias, presentamos algunos resultados preliminares que se aplican en el Capítulo 4,[1],[10],[17].En el Capítulo 4 se presentan algunos resultados conocidos sobre estabilidad en problemas no estacionarios, restringiendo la dinámica a una variedad central.…”