Abstract
A dynamical characterization of the stability boundary for a fairly largeclass of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddlenode equilibrium points on the stability boundary. The stability boundary of anasymptotically stable equilibrium point is shown to consist of the stable manifoldsof the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.
Highlights
Stable equilibrium points of many practical nonlinear dynamical systems are not globally stable
This paper developed the theory of stability region of nonlinear dynamical systems by generalizing the existing results on the characterization of the stability boundary of asymptotically stable equilibrium points
The generalization developed in this paper considers the existence of a particular type of non-hyperbolic equilibrium point on the stability boundary, the so called saddle-node equilibrium point
Summary
Stable equilibrium points of many practical nonlinear dynamical systems are not globally stable. The exact stability region is of difficult determination and, over the last thirty years, a great number of methods were proposed for estimating the stability region of attractors of nonlinear dynamical systems [16]. The stability boundary of an asymptotically stable equilibrium point was characterized in terms of the stable manifolds of a set of unstable equilibria (and/or closed orbits) lying on this boundary [6] These existing characterizations of stability boundaries were proved under the key assumption that all the equilibrium points on the stability boundary are hyperbolic. Necessary and sufficient conditions for a saddle-node equilibrium point lying on the stability boundary are developed
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