  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.
