Estimation of the region of attraction for nonlinear autonomous systems

We study on the determination of the estimations for the regions of attraction of the nonlinear autonomous systems with polynomial vector fields. The aim is to obtain estimations for the regions of attraction of specific examples of nonlinear autonomous systems by using different methods . The study begins with a review of the Lyapunov stability theory and LaSalle's extension principle of that theory. We introduce dynamical properties of the specific examples of nonlinear systems under consideration. We present the conditions for the characterization of the boundary of region of attraction, and state and prove theorems on the unboundedness of the regions of attraction of the examples. The estimations for the regions of attraction are determined as subsets of those regions by applying Lyapunov methods while the estimations for regions of attraction and also for their boundaries are computed by using non-Lyapunov methods. When implementing these method,s we either modify some of the given algorithms in the literature or state new algorithms instead of using them.