Stability regions of non-hyperbolic dynamical systems: theory and optimal estimation

Abstract

The concept of a stability region (region of attraction) of nonlinear dynamical systems is widely used in many fields such as engineering and the sciences. In this paper, we study the notion of stability regions for a class of non-hyperbolic dynamical systems. A complete characterization of the stability boundary is presented for a fairly large class of non-hyperbolic dynamical systems. Several necessary and sufficient conditions for an equilibrium manifold (the generalized concept of an equilibrium point) to lie on the stability boundary are derived. It is shown that the stability boundary of this class of systems consists of the union of the stable manifolds of the equilibrium manifolds on the stability boundary. In addition, an effective scheme to estimate stability regions by using an energy function is developed. It is shown that the scheme can optimally estimate stability regions for a class of non-hyperbolic dynamical systems. Two examples are given to illustrate the theoretical prediction.