Characterization of saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical system

Abstract

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddle-node equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-zero saddle-node equilibrium points on the stability boundary and the stable center and center manifolds of the type-k saddle-node equilibrium points with k≥ 1 on the stability boundary.