A computational approach for studying domains of attraction for non-linear systems

Abstract

The determination of the domains of attraction of equilibrium states and of periodic solutions plays an important role in delineating the global behavior of non-linear dynamical systems. In this paper, we propose a computational scheme of backward mappings for obtaining the global domains of attraction. The approach is based on the Lyapunov stability theory and extends a method proposed by C.S. Hsu for discrete time systems. The method consists of an iterative procedure to be applied to asymptotically stable equilibrium states and stable periodic solutions and converges to the total domain of attraction of the given solution. The computational scheme proposed here is applicable to both time-invariant and periodic, continuous-time dynamical systems as well as discrete-time dynamical systems. In the paper the mathematical basis for the approach and the computational algorithm are described. The usefulness of the method is illustrated by applying it to different classes of dynamical systems. The study indicates that the method is efficient in analyzing the global behavior of multidimensional systems, both time-invariant and time varying.