Parametric dependence of large disturbance response and relationship to stability boundary

Abstract

This paper considers a system of ordinary differential equations subject to a parameter-dependent disturbance. The goal is to find the boundary in parameter space between parameter values for which the system will recover from the disturbance to a desired stable equilibrium point, and parameter values for which it will not recover. If the system state when the disturbance clears, call it the initial condition, depends continuously on parameter value, then it seems plausible that this parameter space boundary would consist of parameter values whose corresponding initial conditions lie on the boundary of the region of attraction (RoA) of the desired stable equilibrium point (SEP). Unfortunately, this is not true in general since, even when the system's vector field varies smoothly with parameter value, the boundary of the RoA of the SEP may not vary even continuously with respect to small parameter variations. This work shows that, for a large class of vector fields which generalize Morse-Smale vector fields, the RoA boundary varies continuously in an appropriate sense with respect to small parameter variations. Furthermore, it has been shown elsewhere that the RoA boundary for these vector fields is equal to the union of the stable manifolds of the equilibria and periodic orbits they contain. A complete argument is provided here that this decomposition into stable manifolds persists under small changes in parameter for the vector fields under consideration. The above results are applied to provide a theoretical basis for a numerical algorithm which computes parameter values which lie on the desired parameter space boundary.