On the determination of the basin of attraction of discrete dynamical systems

Consider a discrete dynamical system given by the iteration x(n+1) = g(x(n)) with exponentially asymptotically stable fixed point (x) over bar. In this paper, we seek to study its basin of attraction A((x) over bar) using sublevel sets of Lyapunov functions. We prove the existence of a smooth Lyapunov function. Moreover, we present an approximation method of this Lyapunov function using radial basis functions. Error estimates show that one can determine every connected and bounded subset of the basin of attraction with this method. Examples include an application to the region of convergence of Newton's method.