Approximation of the domain of attraction of an asymptotically stable fixed point of a first order analytical system of difference equations

In this paper a first order analytical system of difference equations is considered. For an asymptotically stable fixed point x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when the matrix of the linearized system in x0 is a contraction. This technique is based on the gradual extension of the "embryo" of an analytic function of several variables. The analytic function is a Lyapunov function whose natural domain of analyticity is the DA and which satisfies an iterative functional equation. The equation permits to establish an "embryo" of the Lyapunov function and a first approximation of the DA. The "embryo" is used for the determination of a new "embryo" and a new part of the DA. In this way, computing new "embryos" and new domains, the DA is gradually approximated. Numerical examples are given for polynomial systems.